// Numbas version: exam_results_page_options {"name": "Laws of Indices", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "preamble": {"js": "", "css": ""}, "advice": "

#### a)

\n

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

\n

Using this rule,

\n

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

\n

#### b)

\n

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

\n

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

\n

To do this, write the product out explicitly, as

\n

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

\n

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

\n

Therefore:

\n

\\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}

\n

#### c)

\n

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

\n

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

\n

To do this, we just have to use the previously mentioned rule of indices. We write this out explicity as

\n

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

\n

Using rules of indices,

\n

\\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}

\n

Therefore,

\n

\\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}

\n

Alternatively,

\n

Using the rule of indices: $a^{-m} = \\displaystyle\\frac{1}{a^{m}}$, we can rewrite the question as:

\n

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

\n

And then using the rule: $a^m \\times a^n = a^{m+n}$, this becomes:

\n

\\begin{align}
\\simplify{{b}/{g}} \\times a^\\var{x} \\times a^{-\\var{y}} &= \\simplify{{b}/{g}} \\times a^\\simplify[all,!collectNumbers]{{x}+(-{y})}\\\\
&= \\simplify{{b}/{g}*a^{x-y}}.
\\end{align}

\n

#### d)

\n

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

\n

To do this we use the rules:

\n

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

\n

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

\n

We can then expand the equation as

\n

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

\n

Then using the rule of indices mentioned previously,

\n

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

\n

#### e)

\n

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

\n

To do this we use the rules:

\n

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

\n

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

\n

We can expand the expression as follows:

\n

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

\n

#### f)

\n

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

\n

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

\n

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

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

Used in parts b and e

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

Used in parts b,d and f

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

Used in parts a,c and e

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

\n

Used in parts a,c and f

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

Used in part c

", "templateType": "anything"}, "c": {"name": "c", "definition": "random(2,4)", "group": "Ungrouped variables", "description": "

Used in parts b,d and f

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

Used in part c

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

Used in parts b and d

", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "

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

", "extensions": [], "tags": ["indices", "laws of indices", "powers", "taxonomy"], "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/"}], "ungrouped_variables": ["x", "y", "c", "d", "p", "q", "b", "g"], "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

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

"}, "name": "Laws of Indices", "functions": {}, "parts": [{"customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "customName": "", "stepsPenalty": "1", "steps": [{"customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "prompt": "

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

", "variableReplacements": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "customName": "", "marks": 0, "type": "information", "scripts": {}, "useCustomName": false, "unitTests": []}], "variableReplacements": [], "gaps": [{"checkingType": "absdiff", "customName": "", "checkingAccuracy": 0.001, "answer": "a^{x+y}", "valuegenerators": [{"name": "a", "value": ""}], "checkVariableNames": false, "scripts": {}, "vsetRangePoints": 5, "unitTests": [], "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showCorrectAnswer": true, "mustmatchpattern": {"message": "You haven't simplified: your answer is not in the form $a^?$.", "partialCredit": 0, "nameToCompare": "", "pattern": "a^??"}, "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "failureRate": 1, "type": "jme", "vsetRange": [0, 1], "showPreview": true, "useCustomName": false, "marks": "2"}], "sortAnswers": false, "showCorrectAnswer": true, "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "prompt": "

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

\n

\n

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

", "marks": 0, "type": "gapfill", "scripts": {}, "useCustomName": false, "unitTests": []}, {"customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "customName": "", "variableReplacements": [], "gaps": [{"checkingType": "absdiff", "customName": "", "checkingAccuracy": 0.001, "answer": "{c*d}*a^{p+q}", "valuegenerators": [{"name": "a", "value": ""}], "checkVariableNames": false, "scripts": {}, "vsetRangePoints": 5, "unitTests": [], "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showCorrectAnswer": true, "mustmatchpattern": {"message": "You haven't simplified: your answer is not in the form $?a^?$.", "partialCredit": 0, "nameToCompare": "", "pattern": "$n*a^??"}, "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "failureRate": 1, "type": "jme", "vsetRange": [0, 1], "showPreview": true, "useCustomName": false, "marks": "2"}], "sortAnswers": false, "showCorrectAnswer": true, "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "prompt": " Write$\\var{c}a^\\var{p} \\times \\var{d}a^\\var{q}$as an integer multiplied by a single power of$a$. \n$\\var{c}a^\\var{p} \\times \\var{d}a^\\var{q} =$[[0]]. \n ", "marks": 0, "type": "gapfill", "scripts": {}, "useCustomName": false, "unitTests": []}, {"customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "customName": "", "stepsPenalty": "1", "steps": [{"customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "prompt": " You could use one of the following rules: \n$a^m \\div a^n = a^{m-n}$. \n$a^{-m} = \\displaystyle\\frac{1}{a^m}$. ", "variableReplacements": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "customName": "", "marks": 0, "type": "information", "scripts": {}, "useCustomName": false, "unitTests": []}], "variableReplacements": [], "gaps": [{"checkingType": "absdiff", "customName": "", "checkingAccuracy": 0.001, "answer": "{b}/{g}a^{x-y}", "valuegenerators": [{"name": "a", "value": ""}], "checkVariableNames": false, "scripts": {}, "vsetRangePoints": 5, "unitTests": [], "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showCorrectAnswer": true, "mustmatchpattern": {"message": "You haven't simplified: your answer is not in the form$\\frac{?}{?} \\cdot a^?$.", "partialCredit": 0, "nameToCompare": "", "pattern": "$n/$n? * a^??"}, "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "failureRate": 1, "type": "jme", "vsetRange": [0, 1], "showPreview": true, "useCustomName": false, "marks": "2"}], "sortAnswers": false, "showCorrectAnswer": true, "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "prompt": " Write$\\displaystyle\\simplify{{b}*a^{x}/({g}*a^{y})}$as a number multiplied by a single power of$a$. \n$\\displaystyle\\simplify{{b}*a^{x}/({g}*a^{y})} =$[[0]]. \n ", "marks": 0, "type": "gapfill", "scripts": {}, "useCustomName": false, "unitTests": []}, {"customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "customName": "", "stepsPenalty": 0, "steps": [{"customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "prompt": " Use the rules: \n$(ab)^m = a^mb^m$. \n$(a^m)^n = a^{mn}$. ", "variableReplacements": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "customName": "", "marks": 0, "type": "information", "scripts": {}, "useCustomName": false, "unitTests": []}], "variableReplacements": [], "gaps": [{"checkingType": "absdiff", "customName": "", "checkingAccuracy": 0.001, "answer": "{c^{q}}*a^{p*q}", "valuegenerators": [{"name": "a", "value": ""}], "checkVariableNames": false, "scripts": {}, "vsetRangePoints": 5, "unitTests": [], "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showCorrectAnswer": true, "mustmatchpattern": {"message": "You must write your answer as an integer multiplied by a power of$a$.", "partialCredit": 0, "nameToCompare": "", "pattern": "$n?*a^??"}, "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "failureRate": 1, "type": "jme", "vsetRange": [0, 1], "showPreview": true, "useCustomName": false, "marks": "2"}], "sortAnswers": false, "showCorrectAnswer": true, "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "prompt": "

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

\n

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

\n

", "marks": 0, "type": "gapfill", "scripts": {}, "useCustomName": false, "unitTests": []}, {"customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "customName": "", "stepsPenalty": "1", "steps": [{"customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "prompt": "

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

", "variableReplacements": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "customName": "", "marks": 0, "type": "information", "scripts": {}, "useCustomName": false, "unitTests": []}], "variableReplacements": [], "gaps": [{"checkingType": "absdiff", "customName": "", "checkingAccuracy": 0.001, "answer": "a^(1/{d})", "valuegenerators": [{"name": "a", "value": ""}], "scripts": {}, "checkVariableNames": false, "vsetRangePoints": 5, "unitTests": [], "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "answerSimplification": "basic", "showCorrectAnswer": true, "mustmatchpattern": {"message": "

", "partialCredit": 0, "nameToCompare": "", "pattern": "a^??"}, "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "failureRate": 1, "type": "jme", "vsetRange": [0, 1], "showPreview": true, "useCustomName": false, "marks": "2"}], "sortAnswers": false, "showCorrectAnswer": true, "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "prompt": "

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

\n

$\\sqrt[\\var{d}]{a} =$ [[0]].

", "marks": 0, "type": "gapfill", "scripts": {}, "useCustomName": false, "unitTests": []}, {"customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "customName": "", "stepsPenalty": "1", "steps": [{"customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "prompt": "

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

", "variableReplacements": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "customName": "", "marks": 0, "type": "information", "scripts": {}, "useCustomName": false, "unitTests": []}], "variableReplacements": [], "gaps": [{"checkingType": "absdiff", "customName": "", "checkingAccuracy": 0.001, "answer": "a^({c}/{q})", "valuegenerators": [{"name": "a", "value": ""}], "checkVariableNames": false, "scripts": {}, "vsetRangePoints": 5, "unitTests": [], "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showCorrectAnswer": true, "mustmatchpattern": {"message": "

Write $\\sqrt[\\var{q}]{a^\\var{c}}$ as a single power of $a$.
$\\sqrt[\\var{q}]{a^\\var{c}} =$ [[0]].