// Numbas version: finer_feedback_settings {"name": "Simon's copy of Laws of Indices", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"parts": [{"type": "gapfill", "variableReplacementStrategy": "originalfirst", "gaps": [{"answer": "a^{x+y}", "type": "jme", "variableReplacementStrategy": "originalfirst", "checkingType": "absdiff", "showPreview": true, "checkingAccuracy": 0.001, "scripts": {"mark": {"order": "after", "script": "// Parse the student's answer as a syntax tree\nvar studentTree = Numbas.jme.compile(this.studentAnswer,Numbas.jme.builtinScope);\n\n// Create the pattern to match against \n// we just want two sets of brackets, each containing two terms\n// or one of the brackets might not have a constant term\n// or for repeated roots, you might write (x+a)^2\nvar rule = Numbas.jme.compile('m_any(a^(m_number))');\n\n// Check the student's answer matches the pattern. \nvar m = Numbas.jme.display.matchTree(rule,studentTree,true);\n// If not, take away marks\nif(!m) {\n this.multCredit(0,'You haven\\'t simplified: your answer is not in the form $\\a^?$.');\n}"}}, "extendBaseMarkingAlgorithm": true, "failureRate": 1, "showCorrectAnswer": true, "customMarkingAlgorithm": "", "vsetRangePoints": 5, "notallowed": {"partialCredit": 0, "strings": ["*", "(", ")", "+"], "message": "

You must write your answer as a single power of a.

", "showStrings": true}, "variableReplacements": [], "musthave": {"partialCredit": 0, "strings": ["^"], "message": "

You must write your answer as a single power of a.

", "showStrings": true}, "checkVariableNames": false, "vsetRange": [0, 1], "expectedVariableNames": [], "marks": "2", "unitTests": [], "showFeedbackIcon": true}], "stepsPenalty": "1", "sortAnswers": false, "marks": 0, "scripts": {}, "prompt": "

Write $a^{\\var{x}} \\times a^{\\var{y}}$ as a single power of $a$.

$a^{\\var{x}} \\times a^{\\var{y}} =$ [[0]].

", "showCorrectAnswer": true, "customMarkingAlgorithm": "", "variableReplacements": [], "steps": [{"unitTests": [], "type": "information", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "prompt": "

Use the rule: $a^m \\times a^n = a^{m+n}$.

", "showCorrectAnswer": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "scripts": {}, "showFeedbackIcon": true, "marks": 0}], "extendBaseMarkingAlgorithm": true, "unitTests": [], "showFeedbackIcon": true}, {"type": "gapfill", "variableReplacementStrategy": "originalfirst", "gaps": [{"answer": "{c*d}*a^{p+q}", "type": "jme", "variableReplacementStrategy": "originalfirst", "checkingType": "absdiff", "showPreview": true, "checkingAccuracy": 0.001, "scripts": {"mark": {"order": "after", "script": "// Parse the student's answer as a syntax tree\nvar studentTree = Numbas.jme.compile(this.studentAnswer,Numbas.jme.builtinScope);\n\n// Create the pattern to match against \n// we just want two sets of brackets, each containing two terms\n// or one of the brackets might not have a constant term\n// or for repeated roots, you might write (x+a)^2\nvar rule = Numbas.jme.compile('m_any((m_number)*a^(m_number))');\n\n// Check the student's answer matches the pattern. \nvar m = Numbas.jme.display.matchTree(rule,studentTree,true);\n// If not, take away marks\nif(!m) {\n this.multCredit(0,'You haven\\'t simplified: your answer is not in the form $\\?a^?$.');\n}"}}, "extendBaseMarkingAlgorithm": true, "failureRate": 1, "showCorrectAnswer": true, "customMarkingAlgorithm": "", "vsetRangePoints": 5, "notallowed": {"partialCredit": 0, "strings": ["(", ")"], "message": "

You must write your answer as a number multiplied by a single power of a.

", "showStrings": true}, "variableReplacements": [], "musthave": {"partialCredit": 0, "strings": ["^"], "message": "

You must write your answer as a number multiplied by a single power of a.

", "showStrings": true}, "checkVariableNames": false, "vsetRange": [0, 1], "expectedVariableNames": [], "marks": "2", "unitTests": [], "showFeedbackIcon": true}], "sortAnswers": false, "marks": 0, "scripts": {}, "prompt": "

Write $\\var{c}a^\\var{p} \\times \\var{d}a^\\var{q}$ as an integer multiplied by a single power of $a$.

$\\var{c}a^\\var{p} \\times \\var{d}a^\\var{q} =$ [[0]].

", "showCorrectAnswer": true, "customMarkingAlgorithm": "", "variableReplacements": [], "extendBaseMarkingAlgorithm": true, "unitTests": [], "showFeedbackIcon": true}, {"type": "gapfill", "variableReplacementStrategy": "originalfirst", "gaps": [{"answer": "{b/g}a^{x-y}", "type": "jme", "variableReplacementStrategy": "originalfirst", "checkingType": "absdiff", "showPreview": true, "checkingAccuracy": 0.001, "scripts": {"mark": {"order": "after", "script": "// Parse the student's answer as a syntax tree\nvar studentTree = Numbas.jme.compile(this.studentAnswer,Numbas.jme.builtinScope);\n\n// Create the pattern to match against \n// we just want two sets of brackets, each containing two terms\n// or one of the brackets might not have a constant term\n// or for repeated roots, you might write (x+a)^2\nvar rule = Numbas.jme.compile('m_any( (m_number/m_number)*a^(m_pm(m_number)), (m_number*a^(m_pm(m_number)))/m_number )');\n\n// Check the student's answer matches the pattern. \nvar m = Numbas.jme.display.matchTree(rule,studentTree,true);\n// If not, take away marks\nif(!m) {\n this.multCredit(0,'You haven\\'t simplified: your answer is not in the form $\\\\frac{?}{\\\\cdot}a^n$.');\n}"}}, "extendBaseMarkingAlgorithm": true, "failureRate": 1, "showCorrectAnswer": true, "customMarkingAlgorithm": "", "vsetRangePoints": 5, "variableReplacements": [], "checkVariableNames": false, "vsetRange": [0, 1], "expectedVariableNames": [], "marks": "2", "unitTests": [], "showFeedbackIcon": true}], "stepsPenalty": "1", "sortAnswers": false, "marks": 0, "scripts": {}, "prompt": "

Write $\\displaystyle\\simplify{{b}*a^{x}/({g}*a^{y})}$ as a number multiplied by a single power of $a$.

$\\displaystyle\\simplify{{b}*a^{x}/({g}*a^{y})} =$ [[0]].

You could use one of the following rules:

$a^m \\div a^n = a^{m-n}$.

$a^{-m} = \\displaystyle\\frac{1}{a^m}$.

You must enter your answer as an integer multiplied by a single power of a.

", "showStrings": true}, "variableReplacements": [], "musthave": {"partialCredit": 0, "strings": ["^"], "message": "

You must enter your answer as an integer multiplied by a single power of a.

", "showStrings": true}, "checkVariableNames": false, "vsetRange": [0, 1], "expectedVariableNames": [], "marks": "2", "unitTests": [], "showFeedbackIcon": true}], "stepsPenalty": 0, "sortAnswers": false, "marks": 0, "scripts": {}, "prompt": "

Write $(\\simplify{{c}*a^{p}})^{\\var{q}}$ as an integer multiplied by a single power of $a$.

$(\\simplify{{c}*a^{p}})^{\\var{q}} =$ [[0]].

Use the rules:

$(ab)^m = a^mb^m$.

$(a^m)^n = a^{mn}$.

You must input your answer as a single power of a.

", "showStrings": true}, "musthave": {"partialCredit": 0, "strings": ["^", "(", ")"], "message": "

You must input your answer as a single power of a.

", "showStrings": true}}], "stepsPenalty": "1", "sortAnswers": false, "marks": 0, "scripts": {}, "prompt": "

Write $\\sqrt[\\var{d}]{a}$ as a single power of $a$.

$\\sqrt[\\var{d}]{a} =$ [[0]].

Use the rule: $a^\\frac{1}{m} = \\sqrt[m]{a}$.

You must enter your answer as a single power of a.

", "showStrings": true}, "variableReplacements": [], "musthave": {"partialCredit": 0, "strings": ["^", "(", ")"], "message": "

You must enter your answer as a single power of a.

Write $\\sqrt[\\var{q}]{a^\\var{c}}$ as a single power of $a$.

$\\sqrt[\\var{q}]{a^\\var{c}} =$ [[0]].

Use the rule: $a^\\frac{n}{m} = \\sqrt[m]{a^n}$.

", "showCorrectAnswer": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "scripts": {}, "showFeedbackIcon": true, "marks": 0}], "extendBaseMarkingAlgorithm": true, "unitTests": [], "showFeedbackIcon": true}], "variablesTest": {"condition": "", "maxRuns": 100}, "variable_groups": [], "extensions": [], "variables": {"c": {"definition": "random(2,4)", "name": "c", "description": "

Used in parts b,d and f

", "group": "Ungrouped variables", "templateType": "anything"}, "g": {"definition": "random(2..9 except b)", "name": "g", "description": "

Used in part c

", "group": "Ungrouped variables", "templateType": "anything"}, "x": {"definition": "random(2..9 except y)", "name": "x", "description": "

Used in parts a,c and e

", "group": "Ungrouped variables", "templateType": "anything"}, "q": {"definition": "random(3,5)", "name": "q", "description": "

Used in parts b,d and f

", "group": "Ungrouped variables", "templateType": "anything"}, "y": {"definition": "random(2..6)", "name": "y", "description": "

Used in parts a,c and f

", "group": "Ungrouped variables", "templateType": "anything"}, "b": {"definition": "random(2..9)", "name": "b", "description": "

Used in part c

", "group": "Ungrouped variables", "templateType": "anything"}, "p": {"definition": "random(2..9)", "name": "p", "description": "

Used in parts b and d

", "group": "Ungrouped variables", "templateType": "anything"}, "d": {"definition": "random(3..5)", "name": "d", "description": "

Used in parts b and e

", "group": "Ungrouped variables", "templateType": "anything"}}, "ungrouped_variables": ["x", "y", "c", "d", "p", "q", "b", "g"], "name": "Simon's copy of Laws of Indices", "statement": "

Using the laws of indices, simplify each expression down to its simplest form. Recall that $a^{0} = 1$ for any number $a$.

", "tags": [], "functions": {}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

This question aims to test understanding and ability to use the laws of indices.

"}, "advice": "

a)

Here we are using the rule of indices: $a^m \\times a^n = a^{m+n}$.

Using this rule,

\\[
\\begin{align}
a^\\var{x} \\times a^\\var{y}\\ &= a^\\simplify[all, !collectNumbers]{{x}+{y}}\\\\
&= a^\\var{x+y}.
\\end{align}
\\]

b)

We are asked to find $\\var{c}a^\\var{p} \\times \\var{d}a^\\var{q}$.

Notice there is a constant in front of each of the terms.

To do this, write the product out explicitly, as

\\[\\var{c}a^\\var{p} \\times \\var{d}a^\\var{q} = \\var{c} \\times \\var{d} \\times a^\\var{p} \\times a^\\var{q}.\\]

We know that $\\var{c} \\times \\var{d} = \\var{c*d}$, and using the rule of indices: $a^\\var{p} \\times a^\\var{q} = a^\\var{p+q}$.

Therefore:

\\begin{align}
\\var{c}a^\\var{p} \\times \\var{d}a^\\var{q}&= \\var{c*d} \\times a^\\var{p+q} \\\\
&= \\simplify{{c*d}*a^{p+q}}.
\\end{align}

c)

Here we are using: $a^m \\div a^n = a^{m-n}$.

We are asked to simplify the expression, $\\displaystyle\\simplify{{b}*a^{x}/({g}*a^{y})}$.

To do this, we just have to use the previously mentioned rule of indices.

Start by writing the expression as

\\[\\simplify{{b}*a^{x}/({g}*a^{y})} = \\simplify{{b}/{g}} \\times \\frac{\\simplify{a^{x}}}{{\\simplify{(a^{y})}}}.\\]

Using rules of indices,

\\begin{align} \\frac{a^\\var{x}}{a^\\var{y}} &= a^\\var{x} \\div a^\\var{y}\\\\
&= a^\\simplify[all, !collectNumbers]{{x}-{y}}\\\\
&= a^\\var{x-y}.
\\end{align}

Therefore,

\\begin{align}
\\frac{\\var{b}a^\\var{x}}{\\var{g}a^\\var{y}} &= \\simplify{{b}/{g}} \\times \\simplify{a^{{x}-{y}}}\\\\
&= \\simplify{{b}/{g}*a^{x-y}}.
\\end{align}

d)

The question asks us to simplify $(\\simplify{{c}*a^{p}})^{\\var{q}}$.

To do this we use the rules:

\\[(ab)^m = a^mb^m.\\]

\\[(a^{m})^{n} = a^{mn},\\]

Using the first of these rules we obtain:

\\[(\\simplify{{c}*a^{p}})^{\\var{q}}= \\var{c}^{\\var{q}} \\times (a^{\\var{p}})^{\\var{q}}.\\]

Then using the second rule gives:

\\[
\\begin{align}
(\\simplify{{c}*a^{p}})^{\\var{q}}&= \\simplify{{c}^{q}} \\times a^\\var{p*q}\\\\
&= \\simplify{{c}^{q}*a^{p*q}}.
\\end{align}
\\]

e)

The question asks us to simplify $\\sqrt[\\var{d}]{a}$.

To do this we use the rule:

\\[a^\\frac{1}{m} = \\sqrt[m]{a},\\]

We can expand the expression as follows:

\\[
\\begin{align}
\\sqrt[\\var{d}]{a} &= (\\simplify{a})^\\frac{1}{\\var{d}}\\\\
&= a^\\frac{1}{\\var{d}}.
\\end{align}
\\]

f)

The question requires us to simplify $\\sqrt[\\var{c}]{a^\\var{q}}$.

Here, we use the rule of indices: $a^\\frac{n}{m} = \\sqrt[m]{a^n}$, allowing us to expand the expression as follows:

\\[
\\begin{align}
\\sqrt[\\var{c}]{\\simplify{a^{q}}} &= \\simplify[fractionnumbers,all]{(a^{q})^{{1}/{{c}}}}\\\\
&= \\simplify[fractionnumbers,all]{a^{{q}/{c}}}.
\\end{align}
\\]

", "preamble": {"js": "", "css": ""}, "rulesets": {}, "type": "question", "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Elliott Fletcher", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1591/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Elliott Fletcher", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1591/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}