// Numbas version: exam_results_page_options
{"name": "Jinhua's copy of Indices 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["a", "b", "c", "d", "f", "g", "h", "j", "k", "l", "f1", "f2", "f3", "f4", "t1", "t2"], "name": "Jinhua's copy of Indices 2", "tags": [], "preamble": {"css": "", "js": ""}, "advice": "<p><em><strong>Recall the laws of indices to help solve the problems:</strong></em></p>\n<p>$x^a \\times x^b = x^{a+b}$</p>\n<p>$x^a \\div x^b = x^{a-b}$</p>\n<p>$x^{-a} = \\frac{1}{x^a}$</p>\n<p>$(x^a)^b = x^{ab}$</p>\n<p>$(\\frac{x}{y})^a = \\frac{x^a}{y^a}$</p>\n<p>$x^\\frac{a}{b} = \\sqrt[b]{x^a}$</p>\n<p>$x^0 = 1$</p>\n<p>&nbsp;</p>\n<p><em><strong>Worked Solutions:</strong></em></p>\n<p><em></em>Part a)</p>\n<p>Negative indices are used in the same way as positive&nbsp;indices, except that the&nbsp;index is applied to the reciprocal of the base. That is, the&nbsp;fraction which represents the base value&nbsp;is flipped upside-down.</p>\n<p>In this case, $\\var{c}$, which&nbsp;is represented&nbsp;by the fraction&nbsp;$\\frac{\\var{c}}{1}$, is raised to the power of $\\simplify{1/-{b}}$. This can be simplified in the following way:</p>\n<p>$\\frac{\\var{c}}{1}^\\simplify{1/-{b}}=\\frac{\\var{1}}{\\var{c}}^\\simplify{1/{b}}$</p>\n<p>You can then follow&nbsp;the more familiar process of evaluating positive powers:</p>\n<p>$\\frac{1}{\\var{c}}^\\simplify{1/{b}}=\\frac{1}{{\\var{c}}^\\simplify{1/{b}}}$<span data-jme-visible=\"b=3 or b=4\">$=\\frac{1}{\\sqrt[{\\var{b}}]{\\var{c}}}$</span><span data-jme-visible=\"b=2\">$=\\frac{1}{\\sqrt{4}}=$</span>$=\\frac{1}{2}$</p>\n<p>&nbsp;</p>\n<p>Part f)</p>\n<p>In this case, the base is already presented in fraction form. Usingthe steps decribed above, it can be simplified as follows:</p>\n<p>$\\left(\\simplify{3/11}\\right)^{-2}=\\left(\\simplify{11/3}\\right)^{2}=\\left(\\frac{11^2}{3^2}\\right)=\\left(\\frac{\\simplify{11^2}}{\\simplify{3^2}}\\right)$</p>\n<p></p>\n<p>Part g)</p>\n<p>$\\left(\\simplify{1/{f4}}\\right)^{\\simplify{-{f2}/{f1}}}=\\left(\\frac{\\var{f4}}{1}\\right)^{\\simplify{{f2}/{f1}}}=\\var{f4}^{\\simplify{{f2}/{f1}}}=\\simplify{{f4}^{{f2}/{f1}}}$</p>", "rulesets": {}, "parts": [{"prompt": "<p>$\\var{c}^{\\simplify{-1/{b}}}$</p>", "allowFractions": true, "variableReplacements": [], "maxValue": "1/2", "minValue": "1/2", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": true, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"prompt": "<p>$\\var{d}^{\\simplify{-1/{a}}}$</p>", "allowFractions": true, "variableReplacements": [], "maxValue": "1/3", "minValue": "1/3", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": true, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"prompt": "<p>$\\var{h}^{\\simplify{-{f}/{g}}}$</p>", "allowFractions": true, "variableReplacements": [], "maxValue": "3^-{f}", "minValue": "3^-{f}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": true, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"prompt": "<p>$\\var{l}^{\\simplify{-{k}/{j}}}$</p>", "allowFractions": true, "variableReplacements": [], "maxValue": "2^-k", "minValue": "2^-k", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": true, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"prompt": "<p>$\\var{f3}^{\\simplify{-{f1}/{f2}}}$</p>", "allowFractions": true, "variableReplacements": [], "maxValue": "5^-f1", "minValue": "5^-f1", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": true, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"prompt": "<p>$(\\simplify{3/11})^{-2}$</p>", "allowFractions": true, "variableReplacements": [], "maxValue": "121/9", "minValue": "121/9", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": true, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"prompt": "<p>$(\\simplify{1/{f4}})^{\\simplify{-{f2}/{f1}}}$</p>", "allowFractions": true, "variableReplacements": [], "maxValue": "5^{f2}", "minValue": "5^{f2}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": true, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"prompt": "<p>$(\\simplify{{t1}/{t2}})^{\\simplify{{f1}/{f2}}}$</p>", "allowFractions": true, "variableReplacements": [], "maxValue": "(2/3)^{f1}", "minValue": "(2/3)^{f1}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": true, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "extensions": [], "statement": "<p>Simplify the following and find their final values (either in&nbsp;integer or fraction form).</p>\n<p><em><strong>Note:</strong> Look out for the fractions in either the base or the index which&nbsp;may be simplified before proceeding to evaluate. If you're unsure with the method involved, click 'Reveal answers' and study the Advice section. Once you feel comfortable with the theory, click 'Try another question like this one' to regenerate the exercises.</em></p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": "gcd({k},{j})=1"}, "variables": {"a": {"definition": "random(2..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "f1": {"definition": "random(2..4)", "templateType": "anything", "group": "Ungrouped variables", "name": "f1", "description": ""}, "c": {"definition": "2^b", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "random(2..4)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "f4": {"definition": "5^f1", "templateType": "anything", "group": "Ungrouped variables", "name": "f4", "description": ""}, "d": {"definition": "3^a", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "description": ""}, "g": {"definition": "random(2..4 except f)", "templateType": "anything", "group": "Ungrouped variables", "name": "g", "description": ""}, "f": {"definition": "random(2..4)", "templateType": "anything", "group": "Ungrouped variables", "name": "f", "description": ""}, "h": {"definition": "3^g", "templateType": "anything", "group": "Ungrouped variables", "name": "h", "description": ""}, "k": {"definition": "random(2..5 except j)", "templateType": "anything", "group": "Ungrouped variables", "name": "k", "description": ""}, "j": {"definition": "random(2..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "j", "description": ""}, "t2": {"definition": "3^f2", "templateType": "anything", "group": "Ungrouped variables", "name": "t2", "description": ""}, "l": {"definition": "2^j", "templateType": "anything", "group": "Ungrouped variables", "name": "l", "description": ""}, "f2": {"definition": "random(2..4 except f1)", "templateType": "anything", "group": "Ungrouped variables", "name": "f2", "description": ""}, "t1": {"definition": "2^f2", "templateType": "anything", "group": "Ungrouped variables", "name": "t1", "description": ""}, "f3": {"definition": "5^f2", "templateType": "anything", "group": "Ungrouped variables", "name": "f3", "description": ""}}, "metadata": {"description": "<p>Simplifying indices.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Jinhua Mathias", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/514/"}]}]}], "contributors": [{"name": "Jinhua Mathias", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/514/"}]}