// Numbas version: finer_feedback_settings
{"feedback": {"allowrevealanswer": true, "showactualmark": true, "showanswerstate": true, "advicethreshold": 0, "intro": "", "feedbackmessages": [], "showtotalmark": true, "enterreviewmodeimmediately": true, "showexpectedanswerswhen": "inreview", "showpartfeedbackmessageswhen": "always", "showactualmarkwhen": "always", "showtotalmarkwhen": "always", "showanswerstatewhen": "always", "showadvicewhen": "never"}, "duration": 0, "showQuestionGroupNames": false, "showstudentname": true, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Questions on powers, the laws of indices, and exponential growth.</p>"}, "navigation": {"showfrontpage": true, "allowregen": true, "browse": true, "preventleave": true, "onleave": {"message": "", "action": "none"}, "showresultspage": "oncompletion", "reverse": true}, "question_groups": [{"pickQuestions": 1, "pickingStrategy": "all-ordered", "name": "Group", "questions": [{"name": "Calculate powers of ten", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Stanislav Duris", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1590/"}], "advice": "<p>When $n$ is <strong>positive</strong>, we multiply $10$ by itself $n$ times,</p>\n<p style=\"padding-left: 30px;\">\\[\\text{e.g. } 10^3 =&nbsp;10 \\times 10 \\times 10 = 1000 \\text{ .}\\]</p>\n<p>When $n$ is <strong>negative</strong>, we can think of $10^{-n}$ as $\\frac{1}{10^{n}}$,</p>\n<p style=\"padding-left: 30px;\">\\[\\text{e.g. } 10^{-3} =&nbsp;\\frac{1}{10^3} = \\frac{1}{1000} = 0.001\\text{ .}\\]</p>\n<p>When $n = 0$:</p>\n<p style=\"padding-left: 30px;\">\\[10^{0} = 1 \\text{ .}\\]</p>\n<p>Generally, we can think of $10^n$ as a number in standard form $1 \\times 10^n$. Then $n$ always tells us the number of decimal places to move the decimal point in $1.0$, for example</p>\n<p style=\"padding-left: 30px;\">\\[10^{-3} = 1.0 \\times 10^{-3} \\text{ and since } n = - 3 \\text{, we go } 3 \\text{ places back as follows: } 1.0 &rArr; 0.1 &rArr; 0.01 &rArr; 0.001 \\text{ .}\\]</p>\n<p>A complete table of powers of ten for $n$ from $-6$ to $6$ is:&nbsp;</p>\n<table style=\"border-color: #000000;\" border=\"1\">\n<tbody>\n<tr>\n<th style=\"text-align: center; vertical-align: middle; background-color: #d6d6d6;\">$n$</th>\n<th style=\"text-align: center; vertical-align: middle; background-color: #d6d6d6;\">$10^n$</th>\n</tr>\n<tr>\n<td style=\"height: 50px; text-align: center; vertical-align: middle;\">$-6$</td>\n<td style=\"width: 200px; height: 50px; text-align: center; vertical-align: middle;\">$0.000001$</td>\n</tr>\n<tr>\n<td style=\"height: 50px; text-align: center; vertical-align: middle;\">$-5$</td>\n<td style=\"width: 200px; height: 50px; text-align: center; vertical-align: middle;\">$0.00001$</td>\n</tr>\n<tr>\n<td style=\"height: 50px; text-align: center; vertical-align: middle;\">$-4$</td>\n<td style=\"width: 200px; height: 50px; text-align: center; vertical-align: middle;\">$0.0001$</td>\n</tr>\n<tr>\n<td style=\"height: 50px; text-align: center; vertical-align: middle;\">$-3$</td>\n<td style=\"width: 200px; height: 50px; text-align: center; vertical-align: middle;\">$0.001$</td>\n</tr>\n<tr>\n<td style=\"height: 50px; text-align: center; vertical-align: middle;\">$-2$</td>\n<td style=\"width: 200px; height: 50px; text-align: center; vertical-align: middle;\">$0.01$</td>\n</tr>\n<tr>\n<td style=\"height: 50px; text-align: center; vertical-align: middle;\">$-1$</td>\n<td style=\"width: 200px; height: 50px; text-align: center; vertical-align: middle;\">$0.1$</td>\n</tr>\n<tr>\n<td style=\"height: 50px; text-align: center; vertical-align: middle;\">$0$</td>\n<td style=\"width: 200px; height: 50px; text-align: center; vertical-align: middle;\">$1$</td>\n</tr>\n<tr>\n<td style=\"height: 50px; text-align: center; vertical-align: middle;\">$1$</td>\n<td style=\"width: 200px; height: 50px; text-align: center; vertical-align: middle;\">$10$</td>\n</tr>\n<tr>\n<td style=\"height: 50px; text-align: center; vertical-align: middle;\">$2$</td>\n<td style=\"width: 200px; height: 50px; text-align: center; vertical-align: middle;\">$100$</td>\n</tr>\n<tr>\n<td style=\"height: 50px; text-align: center; vertical-align: middle;\">$3$</td>\n<td style=\"width: 200px; height: 50px; text-align: center; vertical-align: middle;\">$1000$</td>\n</tr>\n<tr>\n<td style=\"height: 50px; text-align: center; vertical-align: middle;\">$4$</td>\n<td style=\"width: 200px; height: 50px; text-align: center; vertical-align: middle;\">$10000$</td>\n</tr>\n<tr>\n<td style=\"height: 50px; text-align: center; vertical-align: middle;\">$5$</td>\n<td style=\"width: 200px; height: 50px; text-align: center; vertical-align: middle;\">$100000$</td>\n</tr>\n<tr>\n<td style=\"height: 50px; text-align: center; vertical-align: middle;\">$6$</td>\n<td style=\"width: 200px; height: 50px; text-align: center; vertical-align: middle;\">$1000000$</td>\n</tr>\n</tbody>\n</table>\n<p></p>", "statement": "<p>Powers of ten can be useful while working with standard index numbers. Fill in the following table of powers of ten:</p>", "variables": {"n": {"name": "n", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "random(2..5)"}}, "tags": ["powers", "powers of 10", "standard index numbers", "taxonomy"], "ungrouped_variables": ["n"], "functions": {}, "preamble": {"js": "", "css": ""}, "type": "question", "variable_groups": [], "rulesets": {}, "variablesTest": {"condition": "", "maxRuns": "1000"}, "metadata": {"description": "<p>Fill in a table of powers of 10.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "parts": [{"scripts": {}, "showCorrectAnswer": true, "gaps": [{"notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "variableReplacements": [], "mustBeReducedPC": 0, "minValue": "{-n-1}", "correctAnswerFraction": false, "allowFractions": false, "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "scripts": {}, "maxValue": "{-n-1}", "showCorrectAnswer": true, "type": "numberentry", "marks": 1, "showFeedbackIcon": true}, {"notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "variableReplacements": [], "mustBeReducedPC": 0, "minValue": "10^{-n+1}", "correctAnswerFraction": false, "allowFractions": false, "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "scripts": {}, "maxValue": "10^{-n+1}", "showCorrectAnswer": true, "type": "numberentry", "marks": 1, "showFeedbackIcon": true}, {"notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "variableReplacements": [], "mustBeReducedPC": 0, "minValue": "10^{n}", "correctAnswerFraction": false, "allowFractions": false, "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "scripts": {}, "maxValue": "10^{n}", "showCorrectAnswer": true, "type": "numberentry", "marks": 1, "showFeedbackIcon": true}, {"notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "variableReplacements": [], "mustBeReducedPC": 0, "minValue": "{n+2}", "correctAnswerFraction": false, "allowFractions": false, "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "scripts": {}, "maxValue": "{n+2}", "showCorrectAnswer": true, "type": "numberentry", "marks": 1, "showFeedbackIcon": true}], "type": "gapfill", "prompt": "<p></p>\n<table style=\"border-color: #000000;\" border=\"1\">\n<tbody>\n<tr>\n<th style=\"text-align: center; vertical-align: middle; background-color: #d6d6d6;\">$n$</th>\n<th style=\"text-align: center; vertical-align: middle; background-color: #d6d6d6;\">$10^n$</th>\n</tr>\n<tr>\n<td style=\"height: 50px; text-align: center; vertical-align: middle;\">[[0]]</td>\n<td style=\"width: 200px; height: 50px; text-align: center; vertical-align: middle;\">$\\var{10^(-n-1)}$</td>\n</tr>\n<tr>\n<td style=\"height: 50px; text-align: center; vertical-align: middle;\">$\\var{-n+1}$</td>\n<td style=\"width: 200px; height: 50px; text-align: center; vertical-align: middle;\">[[1]]</td>\n</tr>\n<tr>\n<td style=\"height: 50px; text-align: center; vertical-align: middle;\">$0$</td>\n<td style=\"width: 200px; height: 50px; text-align: center; vertical-align: middle;\">$1$</td>\n</tr>\n<tr>\n<td style=\"height: 50px; text-align: center; vertical-align: middle;\">$1$</td>\n<td style=\"width: 200px; height: 50px; text-align: center; vertical-align: middle;\">$10$</td>\n</tr>\n<tr>\n<td style=\"height: 50px; text-align: center; vertical-align: middle;\">$\\var{n}$</td>\n<td style=\"width: 200px; height: 50px; text-align: center; vertical-align: middle;\">[[2]]</td>\n</tr>\n<tr>\n<td style=\"height: 50px; text-align: center; vertical-align: middle;\">[[3]]</td>\n<td style=\"width: 200px; height: 50px; text-align: center; vertical-align: middle;\">$\\var{10^(n+2)}$</td>\n</tr>\n</tbody>\n</table>\n<p></p>", "marks": 0, "variableReplacements": [], "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst"}]}, {"name": "Square and cube numbers", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Stanislav Duris", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1590/"}], "variable_groups": [], "variables": {"x": {"templateType": "anything", "definition": "sort([x1,x2,x3,x4,x5])", "description": "<p>Sorted list of integers from 1 to 12.</p>", "name": "x", "group": "Ungrouped variables"}, "x1": {"templateType": "anything", "definition": "random(1..3)", "description": "", "name": "x1", "group": "Ungrouped variables"}, "y2": {"templateType": "anything", "definition": "y^2", "description": "<p>Squared number in part c).</p>", "name": "y2", "group": "Ungrouped variables"}, "x5": {"templateType": "anything", "definition": "random(10..12 except x1 except x2 except x3 except x4)", "description": "", "name": "x5", "group": "Ungrouped variables"}, "x4": {"templateType": "anything", "definition": "random(7..10 except x1 except x2 except x3)", "description": "", "name": "x4", "group": "Ungrouped variables"}, "x3": {"templateType": "anything", "definition": "random(5..7 except x1 except x2)", "description": "", "name": "x3", "group": "Ungrouped variables"}, "ly": {"templateType": "anything", "definition": "y2 - 2y + random(2..4)", "description": "<p>Lower bound in part c).</p>", "name": "ly", "group": "Ungrouped variables"}, "x2": {"templateType": "anything", "definition": "random(3..5 except x1)", "description": "", "name": "x2", "group": "Ungrouped variables"}, "uy": {"templateType": "anything", "definition": "y2 + 2y - random(2..5)", "description": "<p>Upper bound in part c).</p>", "name": "uy", "group": "Ungrouped variables"}, "y": {"templateType": "anything", "definition": "random(3..12 except x)", "description": "<p>Answer to part c).</p>", "name": "y", "group": "Ungrouped variables"}}, "type": "question", "parts": [{"showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "scripts": {}, "marks": 0, "showCorrectAnswer": true, "variableReplacements": [], "gaps": [{"correctAnswerFraction": false, "mustBeReduced": false, "type": "numberentry", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "allowFractions": false, "scripts": {}, "minValue": "x[0]*x[0]", "maxValue": "x[0]*x[0]", "marks": 1, "variableReplacements": []}, {"correctAnswerFraction": false, "mustBeReduced": false, "type": "numberentry", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "allowFractions": false, "scripts": {}, "minValue": "x[1]*x[1]", "maxValue": "x[1]*x[1]", "marks": 1, "variableReplacements": []}, {"correctAnswerFraction": false, "mustBeReduced": false, "type": "numberentry", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "allowFractions": false, "scripts": {}, "minValue": "x[2]*x[2]", "maxValue": "x[2]*x[2]", "marks": 1, "variableReplacements": []}, {"correctAnswerFraction": false, "mustBeReduced": false, "type": "numberentry", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "allowFractions": false, "scripts": {}, "minValue": "x[3]*x[3]", "maxValue": "x[3]*x[3]", "marks": 1, "variableReplacements": []}, {"correctAnswerFraction": false, "mustBeReduced": false, "type": "numberentry", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "allowFractions": false, "scripts": {}, "minValue": "x[4]*x[4]", "maxValue": "x[4]*x[4]", "marks": 1, "variableReplacements": []}], "prompt": "<p>Find the following:</p>\n<p>$\\var{x[0]}^2 =$&nbsp; [[0]]</p>\n<p>$\\var{x[1]}^2 =$ [[1]]</p>\n<p>$\\var{x[2]}^2 =$ [[2]]</p>\n<p>$\\var{x[3]}^2 =$ [[3]]</p>\n<p>$\\var{x[4]}^2 =$ [[4]]</p>\n<p></p>"}, {"showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "scripts": {}, "marks": 0, "showCorrectAnswer": true, "variableReplacements": [], "gaps": [{"correctAnswerFraction": false, "mustBeReduced": false, "type": "numberentry", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "allowFractions": false, "scripts": {}, "minValue": "x[0]*x[0]*x[0]", "maxValue": "x[0]*x[0]*x[0]", "marks": 1, "variableReplacements": []}, {"correctAnswerFraction": false, "mustBeReduced": false, "type": "numberentry", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "allowFractions": false, "scripts": {}, "minValue": "x[1]*x[1]*x[1]", "maxValue": "x[1]*x[1]*x[1]", "marks": 1, "variableReplacements": []}, {"correctAnswerFraction": false, "mustBeReduced": false, "type": "numberentry", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "allowFractions": false, "scripts": {}, "minValue": "x[2]*x[2]*x[2]", "maxValue": "x[2]*x[2]*x[2]", "marks": 1, "variableReplacements": []}, {"correctAnswerFraction": false, "mustBeReduced": false, "type": "numberentry", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "allowFractions": false, "scripts": {}, "minValue": "x[3]*x[3]*x[3]", "maxValue": "x[3]*x[3]*x[3]", "marks": 1, "variableReplacements": []}, {"correctAnswerFraction": false, "mustBeReduced": false, "type": "numberentry", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "allowFractions": false, "scripts": {}, "minValue": "x[4]*x[4]*x[4]", "maxValue": "x[4]*x[4]*x[4]", "marks": 1, "variableReplacements": []}], "prompt": "<p>Find the following:</p>\n<p>$\\var{x[0]}^3 =$&nbsp; [[0]]</p>\n<p>$\\var{x[1]}^3 =$ [[1]]</p>\n<p>$\\var{x[2]}^3 =$ [[2]]</p>\n<p>$\\var{x[3]}^3 =$ [[3]]</p>\n<p>$\\var{x[4]}^3 =$ [[4]]</p>"}, {"showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "scripts": {}, "marks": 0, "showCorrectAnswer": true, "variableReplacements": [], "gaps": [{"correctAnswerFraction": false, "mustBeReduced": false, "type": "numberentry", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "allowFractions": false, "scripts": {}, "minValue": "y^2", "maxValue": "y^2", "marks": 1, "variableReplacements": []}, {"correctAnswerFraction": false, "mustBeReduced": false, "type": "numberentry", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "allowFractions": false, "scripts": {}, "minValue": "y", "maxValue": "y", "marks": "1", "variableReplacements": []}], "prompt": "<p>Find a square number $y^2$ between $\\var{ly}$ and $\\var{uy}$ and its integer root $y$.</p>\n<p>$y^2 = $&nbsp; [[0]]</p>\n<p>$y = $&nbsp; [[1]]</p>"}], "advice": "<h4>a)</h4>\n<p>Squared integers are called square numbers. It may be useful to remember the first few square numbers to be able to use them without a calculator.</p>\n<p>Here:</p>\n<p>\\[ \\begin{align} \\var{x[0]}^2 &amp;= \\var{x[0]} \\times \\var{x[0]} \\\\&amp;= \\var{x[0]^2} \\text{.}\\end{align}\\]</p>\n<p>\\[ \\begin{align} \\var{x[1]}^2 &amp;= \\var{x[1]} \\times \\var{x[1]} \\\\&amp;= \\var{x[1]^2} \\text{.}\\end{align}\\]</p>\n<p>\\[ \\begin{align} \\var{x[2]}^2 &amp;= \\var{x[2]} \\times \\var{x[2]} \\\\&amp;= \\var{x[2]^2} \\text{.}\\end{align}\\]</p>\n<p>\\[ \\begin{align} \\var{x[3]}^2 &amp;= \\var{x[3]} \\times \\var{x[3]} \\\\&amp;= \\var{x[3]^2} \\text{.}\\end{align}\\]</p>\n<p>\\[ \\begin{align} \\var{x[4]}^2 &amp;= \\var{x[4]} \\times \\var{x[4]} \\\\&amp;= \\var{x[4]^2} \\text{.}\\end{align}\\]</p>\n<p></p>\n<h4>b)</h4>\n<p>Cubed integers are called cubed numbers. To obtain these, we would typically always&nbsp;use a calculator.</p>\n<p>We can either cube the number $x$:</p>\n<p>\\[ \\begin{align} \\var{x[0]}^3 &amp;= \\var{x[0]} \\times \\var{x[0]} \\times \\var{x[0]} \\\\&amp;= \\var{x[0]^3} \\text{,} \\end{align}\\]</p>\n<p>or we can multiply the square number&nbsp;$(x_n)^2$&nbsp;from part <strong>a) </strong>by the appropriate $x_n$:</p>\n<p>\\[ \\begin{align} \\var{x[0]}^3 &amp;= \\var{x[0]}^2 \\times \\var{x[0]} \\\\&amp;= \\var{x[0]^2} \\times \\var{x[0]}\\\\&amp;= \\var{x[0]^3} \\text{.} \\end{align}\\]</p>\n<p>\\[ \\begin{align} \\var{x[1]}^3 &amp;= \\var{x[1]} \\times \\var{x[1]} \\times \\var{x[1]} \\\\ &amp;= \\var{x[1]^3} \\text{.}\\end{align}\\]</p>\n<p>\\[ \\begin{align} \\var{x[2]}^3 &amp;= \\var{x[2]} \\times \\var{x[2]} \\times \\var{x[2]} \\\\ &amp;= \\var{x[2]^3} \\text{.}\\end{align}\\]</p>\n<p>\\[ \\begin{align} \\var{x[3]}^3 &amp;= \\var{x[3]} \\times \\var{x[3]} \\times \\var{x[3]} \\\\ &amp;= \\var{x[3]^3} \\text{.}\\end{align}\\]</p>\n<p>\\[ \\begin{align} \\var{x[3]}^4 &amp;= \\var{x[4]} \\times \\var{x[4]} \\times \\var{x[4]} \\\\ &amp;= \\var{x[4]^3} \\text{.}\\end{align}\\]</p>\n<p></p>\n<h4>c)</h4>\n<p>Here is&nbsp;a table of square numbers for integers from $1$ to $15$:</p>\n<table style=\"border-color: #000000; width: 933px; height: 100px; margin-right: auto; margin-left: auto;\" cellspacing=\"1\" cellpadding=\"1\" border=\"1\">\n<tbody>\n<tr style=\"height: 11px;\">\n<th style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle; background-color: #d6d6d6;\">$y$</th>\n<td style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle;\">$1$</td>\n<td style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle;\">$2$</td>\n<td style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle;\">$3$</td>\n<td style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle;\">$4$</td>\n<td style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle;\">$5$</td>\n<td style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle;\">$6$</td>\n<td style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle;\">$7$</td>\n<td style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle;\">$8$</td>\n<td style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle;\">$9$</td>\n<td style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle;\">$10$</td>\n<td style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle;\">$11$</td>\n<td style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle;\">$12$</td>\n<td style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle;\">$13$</td>\n<td style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle;\">$14$</td>\n<td style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle;\">$15$</td>\n</tr>\n<tr style=\"height: 1px;\">\n<th style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle; background-color: #d6d6d6;\">$y^2$</th>\n<td style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle;\">$1$</td>\n<td style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle;\">$4$</td>\n<td style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle;\">$9$</td>\n<td style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle;\">$16$</td>\n<td style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle;\">$25$</td>\n<td style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle;\">$36$</td>\n<td style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle;\">$49$</td>\n<td style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle;\">$64$</td>\n<td style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle;\">$81$</td>\n<td style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle;\">$100$</td>\n<td style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle;\">$121$</td>\n<td style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle;\">$144$</td>\n<td style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle;\">$169$</td>\n<td style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle;\">$196$</td>\n<td style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle;\">$225$</td>\n</tr>\n</tbody>\n</table>\n<p>The only square number between $\\var{ly}$&nbsp;and $\\var{uy}$ is $\\var{y2}$.</p>\n<p>To calculate $y$ we must&nbsp;calculate the square root of $y^2$,</p>\n<p>\\[ \\sqrt{\\var{y2}} = \\var{y} \\text{.}\\]</p>\n<p>This is our integer $y$.</p>\n<p></p>\n<p></p>", "tags": ["cube", "indices", "multiplication", "powers", "roots", "square", "taxonomy"], "preamble": {"js": "", "css": ""}, "rulesets": {}, "functions": {}, "ungrouped_variables": ["x1", "x2", "x3", "x4", "x5", "x", "y2", "y", "ly", "uy"], "statement": "<p>Try the following questions on square and cube numbers.</p>", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Find the squares, and cubes, of some numbers.</p>\n<p>Finally, find a square number between two given limits.</p>"}, "variablesTest": {"condition": "", "maxRuns": "1000"}}, {"name": "Always, sometimes or never: square and cube numbers", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Stanislav Duris", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1590/"}], "type": "question", "statement": "<p>For each of the following statements, select one option from \"Always\", \"Sometimes\" or \"Never\".&nbsp;</p>\n<p>Select:</p>\n<ul>\n<li>\"Always\", if it's true for every possible $x$.</li>\n<li>\"Sometimes\", if there&nbsp;is an $x$ for which&nbsp;it's true&nbsp;but also an $x$ for which it's false.</li>\n<li>\"Never\", if&nbsp;there is no&nbsp;$x$ for&nbsp;which it's true.</li>\n</ul>", "variablesTest": {"condition": "", "maxRuns": "1000"}, "variables": {"a": {"group": "Ungrouped variables", "name": "a", "description": "<p>This variable was created solely for the purpose of being able to publish this question.</p>", "templateType": "anything", "definition": "1"}}, "functions": {}, "tags": ["always", "cubic numbers", "never", "sometimes", "square numbers", "taxonomy"], "variable_groups": [], "parts": [{"layout": {"expression": "", "type": "all"}, "scripts": {}, "minMarks": 0, "type": "m_n_x", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "displayType": "radiogroup", "choices": ["<p style=\"text-align: left;\"><em>i)</em> &nbsp;&nbsp;&nbsp;&nbsp; $x^3$ is greater than $x^2$</p>", "<p><em>ii) &nbsp;&nbsp;&nbsp;&nbsp; </em>$x^2$ is greater than $x$</p>", "<p><em>iii) </em>&nbsp;&nbsp;&nbsp; &nbsp;If $x$ is negative, $x^2$ is negative</p>", "<p><em>iv) </em>&nbsp;&nbsp;&nbsp;&nbsp; If $x$ is negative, $x^3$ is negative</p>", "<p><em>v) &nbsp;&nbsp;&nbsp; </em>&nbsp;$x^2 = x$</p>", "<p><em>vi)&nbsp;&nbsp;&nbsp; </em>&nbsp; $x^2 = - x$</p>", "<p><em>vii) </em>&nbsp;&nbsp;&nbsp;&nbsp; $(x+1)^2 \\gt x$</p>", "<p><em>viii)&nbsp;&nbsp;&nbsp; </em>&nbsp; $(x+1)^3 \\gt x$</p>", "<p><em>ix) </em>&nbsp;&nbsp;&nbsp;&nbsp; $x^3 \\times x = x^2 \\times x^2$</p>", "<p><em>x)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </em>$x^2$ has the opposite sign to $x$</p>", "<p><em>xi)</em>&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;$x^3$ has the opposite sign to $x$</p>"], "showFeedbackIcon": true, "answers": ["<p>Always</p>", "<p>Sometimes</p>", "<p>Never</p>"], "warningType": "none", "matrix": [["0", "1", 0], [0, "1", "0"], ["0", 0, "1"], ["1", "0", 0], ["0", "1", 0], ["0", "1", 0], ["1", "0", 0], [0, "1", 0], ["1", 0, "0"], [0, "1", 0], [0, 0, "1"]], "variableReplacements": [], "marks": 0, "maxMarks": 0, "shuffleChoices": false, "maxAnswers": 0, "minAnswers": 0, "shuffleAnswers": false}], "ungrouped_variables": ["a"], "rulesets": {}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Decide whether statements about&nbsp;square and cube numbers are&nbsp;always true, sometimes true or never true.</p>"}, "preamble": {"css": "", "js": ""}, "advice": "<p style=\"text-align: left;\"><em>i)</em></p>\n<p>Suppose $x$ is negative, for example $x = -5$.</p>\n<p>\\[ \\begin{align} x^2 &amp;= (-5)^2 \\\\&amp;= 25 \\\\ x^3 &amp;= (-5)^3 \\\\&amp;= -125 \\\\x^3 &amp;\\lt x^2\\text{.}&nbsp;\\end{align} \\]</p>\n<p>Now let&nbsp;$x$ be positive, for example $x = 5$.</p>\n<p>\\[ \\begin{align} x^2 &amp;= 5^2 \\\\&amp;= 25 \\\\ x^3 &amp;= 5^3 \\\\&amp;= 125 \\\\x^3 &amp;\\gt x^2\\text{.}\\end{align} \\]</p>\n<p>Therefore, $x^3$ is <strong>sometimes</strong> greater than $x^2$.</p>\n<p><em></em></p>\n<p><em></em></p>\n<p><em>ii)</em></p>\n<p>This is true for either $x \\gt 1$ or $x \\lt 0$ but false for&nbsp;$0 \\leq x \\leq 1$. For example, let $x=0.5$. Then</p>\n<p>\\[ \\begin{align} \\text{When&nbsp;} x &amp;= 0.5\\text{,} \\\\x^2 &amp;= 0.25\\text{, so} \\\\ x^2 &amp;\\lt x \\text{.} \\end{align} \\]</p>\n<p>Therefore, $x^2$ is <strong>sometimes</strong> greater than $x$.</p>\n<p><em></em></p>\n<p><em></em></p>\n<p><em>iii)</em></p>\n<p>Multiplying two negative numbers gives a positive answer and multiplying two postive numbers gives a positive answer. Therefore, $x^2$ is <strong>never</strong> negative.</p>\n<p><em></em></p>\n<p><em></em></p>\n<p><em>iv)</em></p>\n<p>Multiplying a negative number by itself an odd number of times always gives a negative answer. For example, let $x = -1$. Then</p>\n<p>\\[\\begin{align} x^3 &amp;= (-1)^3 \\\\&amp;= -1\\times-1\\times-1 \\\\&amp;=1\\times-1 \\\\&amp;=&nbsp;- 1 \\text{.} \\end{align}\\]</p>\n<p>Therefore, if $x$ is negative, $x^3$ is&nbsp;<strong>always </strong>negative.</p>\n<p><em></em></p>\n<p><em></em></p>\n<p><em>v)</em></p>\n<p>This is true when $x = 1$&nbsp;but false for all other values of $x$. Therefore, $x^2$&nbsp;<strong>sometimes</strong> equals&nbsp;$x$.</p>\n<p><em></em></p>\n<p><em></em></p>\n<p><em>vi)</em></p>\n<p>This is true when $x&nbsp;= -1$ but false for all other values of $x$. Therefore,&nbsp;$x^2$<strong>&nbsp;sometimes</strong> equals $-x$.</p>\n<p><em></em></p>\n<p><em></em></p>\n<p><em>vii)</em></p>\n<p>When $x$ is positive, for example $x = 5$:</p>\n<p>\\[ \\begin{align} (x+1)^2 &amp;= (5 + 1)^2 \\\\&amp;=&nbsp;6^2 \\\\&amp;= 36&nbsp;\\gt x = 5&nbsp;\\end{align} \\]</p>\n<p>When $x = 0$:</p>\n<p>\\[ \\begin{align} (x+1)^2 &amp;= (0 + 1)^2 \\\\&amp;=&nbsp;1^2 \\\\&amp;=&nbsp;1&nbsp;\\gt x =&nbsp;0&nbsp;\\end{align} \\]</p>\n<p>When $x$ is negative, such as $x = -4$:</p>\n<p>\\[ \\begin{align} (x+1)^2 &amp;= (-4 + 1)^2 \\\\&amp;=&nbsp;(-3)^2 \\\\&amp;=&nbsp;9&nbsp;\\gt x =&nbsp;-4&nbsp;\\end{align} \\]</p>\n<p>To&nbsp;see the&nbsp;behaviour of $(x+1)^2$ a bit more clearly,&nbsp;we make&nbsp;a table for values $-3 \\leq x \\leq 3$:</p>\n<table style=\"margin-right: auto; margin-left: auto;\" border=\"1\">\n<tbody>\n<tr>\n<th style=\"background-color: #d1d1d1;\"><strong>$x$</strong></th>\n<td style=\"vertical-align: middle;\">$-3$</td>\n<td style=\"vertical-align: middle;\">$-2$</td>\n<td style=\"vertical-align: middle;\">$-1$</td>\n<td style=\"vertical-align: middle;\">$-0.5$</td>\n<td style=\"vertical-align: middle;\">$0$</td>\n<td style=\"vertical-align: middle;\">$0.5$</td>\n<td style=\"vertical-align: middle;\">$1$</td>\n<td style=\"vertical-align: middle;\">$2$</td>\n<td style=\"vertical-align: middle;\">\n<p>$3$</p>\n</td>\n</tr>\n<tr>\n<th style=\"background-color: #d1d1d1;\"><strong>$(x+1)^2$</strong></th>\n<td style=\"vertical-align: middle;\">$4$</td>\n<td style=\"vertical-align: middle;\">$1$</td>\n<td style=\"vertical-align: middle;\">$0$</td>\n<td style=\"vertical-align: middle;\">$0.25$</td>\n<td style=\"vertical-align: middle;\">$1$</td>\n<td style=\"vertical-align: middle;\">$2.25$</td>\n<td style=\"vertical-align: middle;\">$4$</td>\n<td style=\"vertical-align: middle;\">$9$</td>\n<td style=\"vertical-align: middle;\">$16$</td>\n</tr>\n</tbody>\n</table>\n<p>Therefore, $(x+1)^2$ is <strong>always</strong> greater than $x$.</p>\n<p></p>\n<p><em></em></p>\n<p><em>viii)</em></p>\n<p>When $x$ is positive, for example $x = 5$:</p>\n<p>\\[ \\begin{align} (x+1)^3 &amp;= (5 + 1)^3 \\\\&amp;=&nbsp;6^3 \\\\&amp;= 216&nbsp;\\gt x = 5&nbsp;\\end{align} \\]</p>\n<p>When $x = 0$:</p>\n<p>\\[ \\begin{align} (x+1)^3 &amp;= (0 + 1)^3 \\\\&amp;=&nbsp;1^3 \\\\&amp;=&nbsp;1&nbsp;\\gt x =&nbsp;0&nbsp;\\end{align} \\]</p>\n<p>When $x$ is negative, such as $x = -4$:</p>\n<p>\\[ \\begin{align} (x+1)^2 &amp;= (-4 + 1)^3 \\\\&amp;=&nbsp;(-3)^3 \\\\&amp;=&nbsp;-27&nbsp;\\lt x =&nbsp;-4&nbsp;\\end{align} \\]</p>\n<p>To&nbsp;see the&nbsp;behaviour of $(x+1)^3$ a bit more clearly,&nbsp;we make a table for values $-3 \\leq x \\leq 3$:</p>\n<table style=\"margin-right: auto; margin-left: auto;\" border=\"1\">\n<tbody>\n<tr>\n<th style=\"background-color: #d1d1d1;\"><strong>$x$</strong></th>\n<td style=\"vertical-align: middle;\">$-3$</td>\n<td style=\"vertical-align: middle;\">$-2$</td>\n<td style=\"vertical-align: middle;\">$-1$</td>\n<td style=\"vertical-align: middle;\">$-0.5$</td>\n<td style=\"vertical-align: middle;\">$0$</td>\n<td style=\"vertical-align: middle;\">$0.5$</td>\n<td style=\"vertical-align: middle;\">$1$</td>\n<td style=\"vertical-align: middle;\">$2$</td>\n<td style=\"vertical-align: middle;\">\n<p>$3$</p>\n</td>\n</tr>\n<tr>\n<th style=\"background-color: #d1d1d1;\"><strong>$(x+1)^3$</strong></th>\n<td style=\"vertical-align: middle;\">$-8$</td>\n<td style=\"vertical-align: middle;\">$-1$</td>\n<td style=\"vertical-align: middle;\">$0$</td>\n<td style=\"vertical-align: middle;\">$0.125$</td>\n<td style=\"vertical-align: middle;\">$1$</td>\n<td style=\"vertical-align: middle;\">$3.375$</td>\n<td style=\"vertical-align: middle;\">$8$</td>\n<td style=\"vertical-align: middle;\">$27$</td>\n<td style=\"vertical-align: middle;\">$64$</td>\n</tr>\n</tbody>\n</table>\n<p>Therefore, $(x+1)^3$ is <strong>sometimes</strong> greater than $x$.</p>\n<p></p>\n<p></p>\n<p><em>ix)</em></p>\n<p>We can write&nbsp;</p>\n<p>\\[x^3\\times x = x \\times x \\times x \\times x = x^2 \\times x^2\\text{.}\\]</p>\n<p>Therefore, $x^3 \\times x$ <strong>always</strong> equals $x^2 \\times x^2$.</p>\n<p></p>\n<p></p>\n<p><em>x)</em></p>\n<p>Since&nbsp;$x^2$ is always positive, $x^2$ only has the opposite sign to $x$ when $x$ is negative.&nbsp;</p>\n<p>Therefore, $x^2$ <strong>sometimes</strong> has the opposite sign to $x$.</p>\n<p></p>\n<p></p>\n<p><em>xi)</em></p>\n<p>As seen in part <em>iv)</em>,&nbsp;multiplying a negative number by itself an odd number of times always gives a negative answer. It is also true that multiplying a positive number by itself an odd number of times will give a positive answer. So, $x^3$ always has the same sign as $x$, since we have an odd power.</p>\n<p>Therefore, $x^3$ <strong>never</strong> has the opposite sign to $x$.</p>\n<p></p>\n<p><em></em></p>"}, {"name": "Laws of Indices", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Chris Graham", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/369/"}, {"name": "Elliott Fletcher", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1591/"}], "tags": ["indices", "laws of indices", "powers", "taxonomy"], "metadata": {"description": "<p>This question aims to test understanding and ability to use the laws of indices.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Using the laws of indices, simplify each expression down to its simplest form. Recall that $a^{0} = 1$ for any number $a$.</p>", "advice": "<h4>a)</h4>\n<p>Here we are using the rule of indices: $a^m \\times a^n = a^{m+n}$.</p>\n<p>Using this rule,&nbsp;</p>\n<p>\\[<br/>\\begin{align}<br/>a^\\var{x} \\times a^\\var{y}\\ &amp;= a^\\simplify[all, !collectNumbers]{{x}+{y}}\\\\<br/>&amp;= a^\\var{x+y}.<br/>\\end{align}<br/>\\]</p>\n<h4>b)</h4>\n<p>We are asked to find $\\var{c}a^\\var{p} \\times \\var{d}a^\\var{q}$.</p>\n<p>Notice there is a constant in front of each of the terms.</p>\n<p>To do this, write the product out explicitly, as</p>\n<p>\\[\\var{c}a^\\var{p} \\times \\var{d}a^\\var{q} = \\var{c} \\times \\var{d} \\times a^\\var{p} \\times a^\\var{q}.\\]</p>\n<p>We know that $\\var{c} \\times \\var{d} = \\var{c*d}$, and using the rule of indices: $a^\\var{p} \\times a^\\var{q} = a^\\var{p+q}$.</p>\n<p>Therefore:</p>\n<p>\\begin{align}<br/>\\var{c}a^\\var{p} \\times \\var{d}a^\\var{q}&amp;= \\var{c*d} \\times a^\\var{p+q} \\\\<br/>&amp;= \\simplify{{c*d}*a^{p+q}}.<br/>\\end{align}</p>\n<h4>c)</h4>\n<p>Here we are using: $a^m \\div a^n = a^{m-n}$.</p>\n<p>We are asked to simplify the expression, $\\displaystyle\\simplify{{b}*a^{x}/({g}*a^{y})}$.</p>\n<p>To do this, we just have to use the previously mentioned rule of indices. We write this out explicity as</p>\n<p>\\[\\simplify{{b}*a^{x}/({g}*a^{y})} =&nbsp;\\simplify{{b}/{g}} \\times \\simplify{a^{x}/(a^{y})}.\\]</p>\n<p>Using rules of indices,</p>\n<p>\\begin{align} &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; \\frac{a^\\var{x}}{a^\\var{y}} &amp;= a^\\var{x} \\div a^\\var{y}\\\\<br/>&amp;= a^\\simplify[all, !collectNumbers]{{x}-{y}}\\\\<br/>&amp;= a^\\var{x-y}.<br/>\\end{align}</p>\n<p>Therefore,</p>\n<p>\\begin{align}<br/>\\frac{\\var{b}a^\\var{x}}{\\var{g}a^\\var{y}} &amp;=&nbsp;\\simplify{{b}/{g}} \\times \\simplify{a^{{x}-{y}}}\\\\<br/> &amp;=&nbsp;\\simplify{{b}/{g}*a^{x-y}}.<br/>\\end{align}</p>\n<p>Alternatively,&nbsp;</p>\n<p>Using the&nbsp;rule of indices: $a^{-m}&nbsp; = \\displaystyle\\frac{1}{a^{m}}$,&nbsp;we can&nbsp;rewrite the question as:</p>\n<p>\\begin{align}<br/>\\frac{\\var{b}a^\\var{x}}{\\var{g}a^\\var{y}} &amp;=&nbsp;\\simplify{{b}/{g}} \\times \\frac{a^\\var{x}}{a^\\var{y}}\\\\<br/> &amp;=&nbsp;\\simplify{{b}/{g}}&nbsp;\\times a^\\var{x} \\times a^{-\\var{y}}.<br/>\\end{align}</p>\n<p>And then using the rule: $a^m \\times a^n = a^{m+n}$, this becomes:</p>\n<p>\\begin{align}<br/>\\simplify{{b}/{g}}&nbsp;\\times a^\\var{x} \\times a^{-\\var{y}} &amp;=&nbsp;\\simplify{{b}/{g}} \\times a^\\simplify[all,!collectNumbers]{{x}+(-{y})}\\\\<br/> &amp;= \\simplify{{b}/{g}*a^{x-y}}.<br/>\\end{align}</p>\n<h4><strong>d) </strong></h4>\n<p>The question asks us to simplify $(\\simplify{{c}*a^{p}})^{\\var{q}}$.</p>\n<p>To do this we use the rules:</p>\n<p>\\[(a^{m})^{n} = a^{mn},\\]</p>\n<p>\\[(ab)^m = a^mb^m.\\]</p>\n<p>We can&nbsp;then&nbsp;expand the equation as</p>\n<p>\\[(\\simplify{{c}*a^{p}})^{\\var{q}}= \\var{c}^{\\var{q}} \\times (a^{\\var{p}})^{\\var{q}}.\\]</p>\n<p>Then using the rule of indices mentioned previously,</p>\n<p>\\[<br/>\\begin{align}<br/>(\\simplify{{c}*a^{p}})^{\\var{q}}&amp;= \\simplify{{c}^{q}} \\times a^\\var{p*q}\\\\<br/> &amp;= \\simplify{{c}^{q}*a^{p*q}}.<br/>\\end{align}<br/>\\]</p>\n<h4>e)</h4>\n<p>The question asks us to simplify $\\sqrt[\\var{d}]{\\var{x}^\\var{d}a}$.</p>\n<p>To do this we use the rules:</p>\n<p>\\[a^\\frac{1}{m} = \\sqrt[m]{a},\\]</p>\n<p>\\[(ab)^m = a^mb^m.\\]</p>\n<p>We can expand the expression as follows:</p>\n<p>\\[<br/>\\begin{align}<br/>\\sqrt[\\var{d}]{a} &amp;=&nbsp;(\\simplify{a})^\\frac{1}{\\var{d}}\\\\<br/> &amp;= a^\\frac{1}{\\var{d}}.<br/>\\end{align}<br/>\\]</p>\n<h4>f)</h4>\n<p>The&nbsp;question requires us to simplify $\\sqrt[\\var{c}]{a^\\var{q}}$.</p>\n<p>Here, we use&nbsp;the rule of indices: $a^\\frac{n}{m} = \\sqrt[m]{a^n}$, allowing us to expand the expression as follows:</p>\n<p>\\[<br/>\\begin{align}<br/>\\sqrt[\\var{c}]{\\simplify{a^{q}}} &amp;= \\simplify[fractionnumbers,all]{(a^{q})^{{1}/{{c}}}}\\\\<br/> &amp;= \\simplify[fractionnumbers,all]{a^{{q}/{c}}}.<br/>\\end{align}<br/>\\]</p>", "rulesets": {}, "variables": {"g": {"name": "g", "group": "Ungrouped variables", "definition": "random(2..9 except b)", "description": "<p>Used in part c</p>", "templateType": "anything"}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(2,4)", "description": "<p>Used in parts b,d and f</p>", "templateType": "anything"}, "x": {"name": "x", "group": "Ungrouped variables", "definition": "random(2..9 except y)", "description": "<p>Used in parts a,c and e</p>", "templateType": "anything"}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(3..5)", "description": "<p>Used in parts b and e</p>", "templateType": "anything"}, "p": {"name": "p", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "<p>Used in parts b and d</p>", "templateType": "anything"}, "q": {"name": "q", "group": "Ungrouped variables", "definition": "random(3,5)", "description": "<p>Used in parts b,d and f</p>", "templateType": "anything"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "<p>Used in part c</p>", "templateType": "anything"}, "y": {"name": "y", "group": "Ungrouped variables", "definition": "random(2..6)", "description": "<p></p>\n<p>Used in parts a,c and f</p>", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["x", "y", "c", "d", "p", "q", "b", "g"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>Write $a^{\\var{x}} \\times a^{\\var{y}}$ as a single power of $a$.</p>\n<p></p>\n<p>$a^{\\var{x}} \\times a^{\\var{y}} =$&nbsp;[[0]].</p>", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>Use the rule:&nbsp;$a^m \\times a^n = a^{m+n}$.</p>"}], "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "2", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "a^{x+y}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "mustmatchpattern": {"pattern": "a^?`?", "partialCredit": 0, "message": "You haven't simplified: your answer is not in the form $a^?$.", "nameToCompare": ""}, "valuegenerators": [{"name": "a", "value": ""}]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>Write $\\var{c}a^\\var{p} \\times \\var{d}a^\\var{q}$ as an integer multiplied by a single power of $a$.</p>\n<p>$\\var{c}a^\\var{p} \\times \\var{d}a^\\var{q} =$&nbsp;[[0]].</p>\n<p></p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "2", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "{c*d}*a^{p+q}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "mustmatchpattern": {"pattern": "$n*a^?`?", "partialCredit": 0, "message": "You haven't simplified: your answer is not in the form $?a^?$.", "nameToCompare": ""}, "valuegenerators": [{"name": "a", "value": ""}]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>Write $\\displaystyle\\simplify{{b}*a^{x}/({g}*a^{y})}$ as a number multiplied by a single power of $a$.</p>\n<p>$\\displaystyle\\simplify{{b}*a^{x}/({g}*a^{y})} =$&nbsp;[[0]].</p>\n<p></p>", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>You could use one of the following rules:</p>\n<p>$a^m \\div a^n = a^{m-n}$.</p>\n<p>$a^{-m} = \\displaystyle\\frac{1}{a^m}$.</p>"}], "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "2", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "{b}/{g}a^{x-y}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "mustmatchpattern": {"pattern": "$n/$n`? * a^?`?", "partialCredit": 0, "message": "You haven't simplified: your answer is not in the form $\\frac{?}{?} \\cdot a^?$.", "nameToCompare": ""}, "valuegenerators": [{"name": "a", "value": ""}]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>Write $(\\simplify{{c}*a^{p}})^{\\var{q}}$ as an integer multiplied by a single power of $a$.</p>\n<p>$(\\simplify{{c}*a^{p}})^{\\var{q}} =$&nbsp;[[0]].</p>\n<p></p>", "stepsPenalty": 0, "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>Use the rules:</p>\n<p>$(ab)^m = a^mb^m$.</p>\n<p>$(a^m)^n = a^{mn}$.</p>"}], "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "2", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "{c^{q}}*a^{p*q}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "mustmatchpattern": {"pattern": "$n`?*a^?`?", "partialCredit": 0, "message": "You must write your answer as an integer multiplied by a power of $a$.", "nameToCompare": ""}, "valuegenerators": [{"name": "a", "value": ""}]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>Write $\\sqrt[\\var{d}]{a}$ as a single power of $a$.&nbsp;</p>\n<p>$\\sqrt[\\var{d}]{a} =$&nbsp;[[0]].</p>", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>Use the rule: $a^\\frac{1}{m} = \\sqrt[m]{a}$.</p>"}], "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "2", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "a^(1/{d})", "answerSimplification": "basic", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "mustmatchpattern": {"pattern": "a^?`?", "partialCredit": 0, "message": "<p>You must input your answer as a single power of a.</p>", "nameToCompare": ""}, "valuegenerators": [{"name": "a", "value": ""}]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>Write $\\sqrt[\\var{q}]{a^\\var{c}}$ as a single power of $a$.</p>\n<p>$\\sqrt[\\var{q}]{a^\\var{c}} =$&nbsp;[[0]].</p>", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>Use the rule: $a^\\frac{n}{m} = \\sqrt[m]{a^n}$.</p>"}], "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "2", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "a^({c}/{q})", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "mustmatchpattern": {"pattern": "a^?`?", "partialCredit": 0, "message": "<p>You must input your answer as a single power of a.</p>", "nameToCompare": ""}, "valuegenerators": [{"name": "a", "value": ""}]}], "sortAnswers": false}], "type": "question"}, {"name": "Working with standard index form ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Stanislav Duris", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1590/"}], "type": "question", "statement": "<p>Working with numbers that are very large or very small can be tricky.</p>\n<p>Standard form allows us to simplify these numbers, using powers of 10.</p>\n<div style=\"text-align: left;\">The standard index form can be defined as</div>\n<div>\\[A&nbsp;\\times 10^n,\\]</div>\n<div>where $1 &le; A &lt; 10$ and $n$&nbsp;is an integer, e.g. $2.26 \\times 10^5$ is a standard form of a number 226000.</div>\n<div>\n<p>&nbsp;</p>\n<p>Write&nbsp;the following in standard index form (for example, for&nbsp;$2.01\\times 10^5$ we would write <code>2.01*10^5</code> in the gap).</p>\n</div>", "variablesTest": {"condition": "", "maxRuns": 100}, "variables": {"int": {"group": "Ungrouped variables", "description": "", "templateType": "anything", "name": "int", "definition": "random(2..9)"}, "A3dp": {"group": "Ungrouped variables", "description": "", "templateType": "anything", "name": "A3dp", "definition": "random(1..10 #0.001)"}, "B": {"group": "Ungrouped variables", "description": "", "templateType": "anything", "name": "B", "definition": "repeat(random(3..9 #0.01), 4)"}, "small5": {"group": "Ungrouped variables", "description": "", "templateType": "anything", "name": "small5", "definition": "random(0.00001..0.0001 #0.00000001)"}, "A5dp": {"group": "Ungrouped variables", "description": "", "templateType": "anything", "name": "A5dp", "definition": "random(1..10 #0.00001)"}, "ran": {"group": "Ungrouped variables", "description": "", "templateType": "anything", "name": "ran", "definition": "random([6,7,8,9,10])"}, "A": {"group": "Ungrouped variables", "description": "", "templateType": "anything", "name": "A", "definition": "repeat(random(1..10 #0.01 except 10), 3)"}}, "functions": {}, "tags": ["conversion", "converting", "standard form", "standard index form", "taxonomy"], "variable_groups": [], "parts": [{"scripts": {}, "variableReplacements": [], "marks": 0, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "gaps": [{"checkingtype": "absdiff", "scripts": {}, "showpreview": true, "variableReplacementStrategy": "originalfirst", "vsetrangepoints": 5, "showCorrectAnswer": true, "showFeedbackIcon": true, "answersimplification": "!collectnumbers", "checkingaccuracy": 0.001, "expectedvariablenames": [], "marks": "1", "notallowed": {"message": "", "partialCredit": 0, "showStrings": false, "strings": ["^-2", "^(-2)"]}, "vsetrange": [0, 1], "musthave": {"message": "", "partialCredit": 0, "showStrings": false, "strings": ["*10^2"]}, "variableReplacements": [], "answer": "{A[2]}*10^2", "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "prompt": "<p style=\"padding-left: 30px;\">$\\var{A[2]*10^2} = $&nbsp; [[0]]</p>", "type": "gapfill"}, {"scripts": {}, "variableReplacements": [], "marks": 0, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "gaps": [{"checkingtype": "absdiff", "scripts": {}, "showpreview": true, "variableReplacementStrategy": "originalfirst", "vsetrangepoints": 5, "showCorrectAnswer": true, "showFeedbackIcon": true, "answersimplification": "!collectnumbers", "checkingaccuracy": 0.001, "expectedvariablenames": [], "marks": "1", "notallowed": {"message": "", "partialCredit": 0, "showStrings": false, "strings": ["*10^1"]}, "vsetrange": [0, 1], "musthave": {"message": "", "partialCredit": 0, "showStrings": false, "strings": ["*10^", "-1"]}, "variableReplacements": [], "answer": "{A3dp}*10^(-1)", "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "prompt": "<p style=\"padding-left: 30px;\">$\\var{A3dp*10^(-1)} = $&nbsp; [[0]]</p>", "type": "gapfill"}, {"scripts": {}, "variableReplacements": [], "marks": 0, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "gaps": [{"checkingtype": "absdiff", "scripts": {}, "showpreview": true, "variableReplacementStrategy": "originalfirst", "vsetrangepoints": 5, "showCorrectAnswer": true, "showFeedbackIcon": true, "answersimplification": "!collectnumbers", "checkingaccuracy": 0.001, "expectedvariablenames": [], "marks": "1", "notallowed": {"message": "", "partialCredit": 0, "showStrings": false, "strings": ["^-7", "^(-7)"]}, "vsetrange": [0, 1], "musthave": {"message": "", "partialCredit": 0, "showStrings": false, "strings": ["*10^7"]}, "variableReplacements": [], "answer": "{A5dp}*10^7", "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "prompt": "<p style=\"padding-left: 30px;\">$\\var{precround(A5dp*10^7,0)} =$&nbsp; [[0]]</p>", "type": "gapfill"}, {"scripts": {}, "variableReplacements": [], "marks": 0, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "gaps": [{"checkingtype": "absdiff", "scripts": {}, "showpreview": true, "variableReplacementStrategy": "originalfirst", "vsetrangepoints": 5, "showCorrectAnswer": true, "showFeedbackIcon": true, "answersimplification": "!collectnumbers", "checkingaccuracy": 0.001, "expectedvariablenames": [], "marks": "1", "notallowed": {"message": "", "partialCredit": 0, "showStrings": false, "strings": ["10^5"]}, "vsetrange": [0, 1], "musthave": {"message": "", "partialCredit": 0, "showStrings": false, "strings": ["*10^", "-5"]}, "variableReplacements": [], "answer": "{{small5}*10^5}*10^(-5)", "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "prompt": "<p style=\"padding-left: 30px;\">$\\var{small5} = $&nbsp; [[0]]</p>", "type": "gapfill"}], "ungrouped_variables": ["A3dp", "A5dp", "small5", "A", "ran", "B", "int"], "rulesets": {}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Convert a variety of numbers from decimal to&nbsp;standard index form.</p>"}, "preamble": {"css": "", "js": ""}, "advice": "<p>Converting from decimal to a standard form, we are looking for $A&nbsp;\\times 10^n$.</p>\n<p>We need make the first number ($A$) between 1 and 10, so we put the decimal place after the first non-zero digit.</p>\n<p>&nbsp;</p>\n<p><strong>a)</strong></p>\n<p>In $\\var{A[2]*10^2}$, the first non-zero digit is $\\var{siground(A[2] - 0.5, 1)}$ so we get $A = \\var{A[2]}$.</p>\n<p style=\"padding-left: 30px;\">If we moved the decimal place&nbsp;in $\\var{A[2]}$ so it matches our original number $\\var{A[2]*10^2}$, we would go 2&nbsp;places to the right, so $n = 2$.</p>\n<p style=\"padding-left: 30px;\">&nbsp; </p>\n<p><strong>b)</strong></p>\n<p>In $\\var{A3dp*10^(-1)}$, the first non-zero digit is $\\var{siground(A3dp - 0.5, 1)}$ so we get&nbsp;$A = \\var{A3dp}$.</p>\n<p style=\"padding-left: 30px;\">If we moved the decimal place&nbsp;in $\\var{A3dp}$ so it matches our original number $\\var{A3dp*10^(-1)}$, we would go&nbsp;1&nbsp;place to the left, so $n = -1$.</p>\n<p style=\"padding-left: 30px;\">&nbsp; </p>\n<p><strong>c)</strong></p>\n<p>In $\\var{precround(A5dp*10^7,0)}$ the first non-zero digit is $\\var{siground(A5dp - 0.5, 1)}$ so we get $A = \\var{A5dp}$.</p>\n<p style=\"padding-left: 30px;\">If we moved the decimal place&nbsp;in $\\var{A5dp}$ so it matches our original number $\\var{precround(A5dp*10^7,0)}$, we would go&nbsp;7&nbsp;places to the right, so $n = 7$.</p>\n<p style=\"padding-left: 30px;\">&nbsp; </p>\n<h4>d)</h4>\n<p>In $\\var{small5}$ the first non-zero digit is {siground({{small5}*10^5} - 0.5, 1)} so we get $A = \\var{small5*10^5}$.</p>\n<p style=\"padding-left: 30px;\">If we moved the decimal place&nbsp;in $\\var{small5*10^5}$ so it matches our original number $\\var{small5}$, we would go&nbsp;5 places to the left, so $n = -5$.</p>\n<p style=\"padding-left: 30px;\"></p>"}, {"name": "Simple interest", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Stanislav Duris", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1590/"}], "metadata": {"description": "<p>Calculate the interest accrued in a savings account, given the initial balance and annual interest rate.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Suppose you put &pound;{money} into a savings account&nbsp;exactly {years} years ago and you haven't touched the money since. The simple interest rate on the account is {perc2}% per year.</p>", "variables": {"perc2": {"name": "perc2", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "random(0..2.5 #0.05)"}, "money": {"name": "money", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "random(100..5000 #100)"}, "years": {"name": "years", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "random(2..6)"}}, "tags": ["interests", "percentages", "taxonomy"], "ungrouped_variables": ["perc2", "money", "years"], "functions": {}, "preamble": {"js": "", "css": ""}, "advice": "<p>This is <em>simple</em> interest, which means the amount added each year is a percentage of the <strong>original</strong> amount. The amount we add is fixed for all {years} years.</p>\n<p>First,&nbsp;we work out the amount of interest for one year:</p>\n<p>\\begin{align} <br/>\\var{perc2} \\text{% of } \\var{money} &amp;= \\frac{\\var{perc2}}{100} \\times \\var{money} \\\\<br/>&amp;= \\var{perc2/100} \\times \\var{money} \\\\<br/>&amp;= &pound;\\var{dpformat(perc2/100*money,2)} \\text{.}&nbsp;<br/>\\end{align}</p>\n<p>The money has been in the account for {years} years, so we multiply $&pound;\\var{dpformat(perc2/100*money,2)}$ by $\\var{years}$.</p>\n<p>\\[ &pound;\\var{dpformat(perc2/100*money,2)} \\times \\var{years} = &pound;\\var{dpformat(perc2/100*money*years,2)} \\text{.} \\]</p>\n<p>Adding this to the original balance:</p>\n<p>\\[ &pound;\\var{money} + &pound;\\var{dpformat(perc2/100*money*years,2)}&nbsp; = &pound;\\var{dpformat(perc2/100*money*years + money,2)} \\text{.} \\]</p>\n<p>This is the amount we would get if we withdrew the whole savings balance today.</p>", "type": "question", "variable_groups": [], "rulesets": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "parts": [{"scripts": {}, "showCorrectAnswer": true, "gaps": [{"notationStyles": ["plain", "en", "si-en"], "precision": "2", "variableReplacements": [], "mustBeReducedPC": 0, "precisionPartialCredit": 0, "minValue": "(1 + perc2/100*years)*money", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "allowFractions": false, "correctAnswerStyle": "plain", "precisionType": "dp", "scripts": {}, "maxValue": "(1 + perc2/100*years)*money", "showCorrectAnswer": false, "strictPrecision": true, "type": "numberentry", "showPrecisionHint": true, "mustBeReduced": false, "marks": "2", "showFeedbackIcon": true, "precisionMessage": "You have not given your answer to the correct precision."}], "type": "gapfill", "marks": 0, "prompt": "<p>&nbsp;If you were to withdraw the money from this account now, how much&nbsp;would you have?</p>\n<p>&pound; [[0]]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true}]}, {"name": "Cumulative percent decrease", "extensions": ["random_person"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Stanislav Duris", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1590/"}, {"name": "Elliott Fletcher", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1591/"}], "type": "question", "tags": ["decrease", "percentages", "taxonomy"], "variablesTest": {"condition": "", "maxRuns": "1000"}, "variables": {"test": {"definition": "precround(precround(price*((100-perc)/100)^5, 2)*((100-perc)/100)^(n-5), 2)", "description": "<p>Calculated value of price2 to ensure we mention rounding&nbsp;errors&nbsp;in advice only when needed.</p>", "templateType": "anything", "name": "test", "group": "Part b)"}, "pricee3": {"definition": "precround(price*((100 - perc)/100)^(testn-2),2)", "description": "", "templateType": "anything", "name": "pricee3", "group": "Part b)"}, "person": {"definition": "random_person()", "description": "", "templateType": "anything", "name": "person", "group": "Part b)"}, "testn": {"definition": "random(6..9)", "description": "<p>Number of months in total.</p>", "templateType": "anything", "name": "testn", "group": "Part b)"}, "pricee1": {"definition": "precround(price*((100 - perc)/100)^(testn),2)", "description": "<p>The resulting price after the&nbsp;total of testn months.</p>", "templateType": "anything", "name": "pricee1", "group": "Part b)"}, "n": {"definition": "if(pricee2 < threshold, testn-1, testn)", "description": "", "templateType": "anything", "name": "n", "group": "Part b)"}, "threshold": {"definition": "siground(pricee1+5,2)", "description": "", "templateType": "anything", "name": "threshold", "group": "Part b)"}, "price": {"definition": "random(300..800) + 0.99", "description": "<p>The original price.</p>", "templateType": "anything", "name": "price", "group": "Part a)"}, "price2": {"definition": "if(pricee2 < threshold, pricee2, pricee1)", "description": "", "templateType": "anything", "name": "price2", "group": "Part b)"}, "perc": {"definition": "random(2..4 #0.5)", "description": "<p>Percentage decrease per month.</p>", "templateType": "anything", "name": "perc", "group": "Part a)"}, "pricee2": {"definition": "precround(price*((100 - perc)/100)^(testn-1),2)", "description": "", "templateType": "anything", "name": "pricee2", "group": "Part b)"}}, "statement": "<p>A smartphone's value decreases by $\\var{perc}$% every month. The original price when it is released is $&pound;\\var{price}$.</p>", "variable_groups": [{"name": "Part a)", "variables": ["price", "perc"]}, {"name": "Part b)", "variables": ["threshold", "pricee1", "pricee2", "pricee3", "testn", "test", "price2", "n", "person"]}], "parts": [{"showCorrectAnswer": true, "scripts": {}, "variableReplacements": [], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "stepsPenalty": 0, "gaps": [{"correctAnswerFraction": false, "precisionMessage": "<p>Round your answer to $2$ decimal places.</p>", "precisionPartialCredit": 0, "scripts": {}, "maxValue": "precround(price*((100-perc)/100)^5, 2)", "variableReplacementStrategy": "originalfirst", "allowFractions": false, "precision": "2", "precisionType": "dp", "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "mustBeReduced": false, "minValue": "precround(price*((100-perc)/100)^5, 2)", "showPrecisionHint": true, "marks": "2", "variableReplacements": [], "strictPrecision": false, "showCorrectAnswer": true, "type": "numberentry"}], "showFeedbackIcon": true, "prompt": "<p>How much&nbsp;will the smartphone&nbsp;be worth after&nbsp;$5$ months?</p>\n<p>&pound; [[0]]</p>", "steps": [{"scripts": {}, "variableReplacements": [], "type": "information", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true, "prompt": "<p>The original price of the phone is $&pound;\\var{price}$ and we are told that the price decreases&nbsp;by $\\var{perc}$% every month.</p>\n<p></p>", "marks": 0}, {"correctAnswerFraction": false, "scripts": {}, "maxValue": "1-{perc}/100", "variableReplacementStrategy": "originalfirst", "allowFractions": false, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "prompt": "<p>What is the decimal multiplier for the decrease in the smartphones each month?</p>", "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "mustBeReduced": false, "minValue": "1-{perc}/100", "variableReplacements": [], "marks": "0.5", "showCorrectAnswer": true, "type": "numberentry"}, {"precisionMessage": "<p>Round your answer to $2$ decimal places.</p>", "precisionPartialCredit": 0, "scripts": {}, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "prompt": "<p>Multiply the original&nbsp;price by the decimal multiplier to obtain the price after 1 month.</p>", "mustBeReducedPC": 0, "mustBeReduced": false, "minValue": "{price}*(1-{perc}/100)", "variableReplacements": [], "marks": "0.5", "strictPrecision": false, "showCorrectAnswer": true, "correctAnswerFraction": false, "notationStyles": ["plain", "en", "si-en"], "allowFractions": false, "precision": "2", "maxValue": "{price}*(1-{perc}/100)", "precisionType": "dp", "correctAnswerStyle": "plain", "showPrecisionHint": true}, {"precisionMessage": "<p>Round your answer to $2$ decimal places.</p>", "precisionPartialCredit": 0, "scripts": {}, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "prompt": "<p>Multiply your answer above&nbsp;by the decimal multiplier to obtain the price after 2 months.&nbsp;</p>\n<p>Note that this is the same as multiplying the original price by $d^2$, where $d$ is the decimal multiplier.</p>", "mustBeReducedPC": 0, "mustBeReduced": false, "minValue": "{price}*(1-{perc}/100)^2", "variableReplacements": [], "marks": "0.5", "strictPrecision": false, "showCorrectAnswer": true, "correctAnswerFraction": false, "notationStyles": ["plain", "en", "si-en"], "allowFractions": false, "precision": "2", "maxValue": "{price}*(1-{perc}/100)^2", "precisionType": "dp", "correctAnswerStyle": "plain", "showPrecisionHint": true}], "marks": 0}, {"scripts": {}, "variableReplacements": [], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "gaps": [{"correctAnswerFraction": false, "scripts": {}, "maxValue": "n-5", "variableReplacementStrategy": "originalfirst", "allowFractions": false, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "mustBeReduced": false, "minValue": "n-5", "variableReplacements": [], "marks": "2", "showCorrectAnswer": true, "type": "numberentry"}], "showFeedbackIcon": true, "prompt": "<p>{person['name']} has&nbsp;$&pound;\\var{threshold}$ to spend on a smartphone. After how many <strong>more</strong>&nbsp;full months&nbsp;will {person['pronouns']['they']}&nbsp;be able to afford the smartphone?</p>\n<p>[[0]] months</p>", "marks": 0}], "ungrouped_variables": [], "rulesets": {}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Given the original price of a smartphone and the rate at which it decreases, calculate its price after a given number of months. In the second part, calculate the time remaining until the price goes below a&nbsp;certain point.</p>"}, "preamble": {"css": "", "js": ""}, "functions": {}, "advice": "<h4>a)</h4>\n<p>We&nbsp;can use the multiplier method to calculate the new price. If the price decreases by {perc}%, this its value is&nbsp;{100-perc}% of the original value after 1 month. The decimal multiplier for {100-perc}% is</p>\n<p>\\[\\frac{\\var{100-perc}}{100} = \\var{(100-perc)/100} \\text{.}\\]</p>\n<p>Each month our smartphone's value can be found by multiplying the previous month's value by the decimal multiplier. For example, after the first month, the value&nbsp;is</p>\n<p>\\[&nbsp;\\var{(100-perc)/100} \\times\\mathrm{&pound;}\\var{price} = \\mathrm{&pound;}\\var{dpformat(price*(100-perc)/100,2)}\\text{.} \\]</p>\n<p>To calculate the price after 5 months, we multiply the original price of the smartphone by our multiplier&nbsp;5 times:</p>\n<p>\\[ \\begin{align}&nbsp;\\text{Final worth} &amp;=&nbsp;\\var{price} \\times \\var{(100-perc)/100}&nbsp;\\times \\var{(100-perc)/100}&nbsp;\\times \\var{(100-perc)/100}&nbsp;\\times \\var{(100-perc)/100}&nbsp;\\times \\var{(100-perc)/100} \\\\&amp;= &nbsp;\\var{price} \\times \\var{(100-perc)/100}^{5} \\\\&amp;= &pound;\\var{precround(price*((100-perc)/100)^5, 2)} {.}&nbsp;\\end{align}\\]</p>\n<h4>b)</h4>\n<p>From part <strong>a)</strong>, the value after 5 months is &nbsp;&pound;$\\var{precround(price*((100-perc)/100)^5, 2)}$.&nbsp;Continuing to multiply the price by the decimal multiplier,</p>\n<p><span data-jme-visible=\"n&gt;5\">\\[&pound;\\var{precround(price*((100-perc)/100)^5, 2)} \\times \\var{(100-perc)/100} = &pound;\\var{precround(precround(price*((100-perc)/100)^5, 2)*(100-perc)/100, 2)}\\]</span></p>\n<p><span data-jme-visible=\"n&gt;6\">\\[&pound;\\var{precround(price*((100-perc)/100)^5, 2)} \\times \\var{(100-perc)/100}^2 = &pound;\\var{precround(precround(price*((100-perc)/100)^6, 2)*(100-perc)/100, 2)}\\]</span></p>\n<p><span data-jme-visible=\"n&gt;7\">\\[&pound;\\var{precround(price*((100-perc)/100)^5, 2)} \\times \\var{(100-perc)/100}^3 = &pound;\\var{precround(precround(price*((100-perc)/100)^7, 2)*(100-perc)/100, 2)}\\]</span></p>\n<p><span data-jme-visible=\"n&gt;8\">\\[&pound;\\var{precround(price*((100-perc)/100)^5, 2)} \\times \\var{(100-perc)/100}^4 = &pound;\\var{precround(precround(price*((100-perc)/100)^8, 2)*(100-perc)/100, 2)}\\]</span></p>\n<p></p>\n<p>The smartphone's value&nbsp;will be below $&pound;\\var{threshold}$ after&nbsp;<strong>{n-5}</strong> more&nbsp;month<span data-jme-visible=\"n&gt;6\">s</span> ({n} months in total since&nbsp;its release).</p>"}]}], "timing": {"timedwarning": {"message": "", "action": "none"}, "allowPause": true, "timeout": {"message": "", "action": "none"}}, "name": "Maria's copy of Powers and indices", "percentPass": 0, "type": "exam", "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Maria Aneiros", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3388/"}], "extensions": ["random_person"], "custom_part_types": [], "resources": []}