// Numbas version: finer_feedback_settings {"name": "Ed'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": [{"statement": "

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

", "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. We write this out explicity as

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

Alternatively,

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

\\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}

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

\\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}

d)

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

To do this we use the rules:

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

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

We can then expand the equation as

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

Then using the rule of indices mentioned previously,

\\[
\\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 $\\sqrt[\\var{d}]{\\var{x}^\\var{d}a}$.

To do this we use the rules:

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

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

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}
\\]

", "tags": ["indices", "laws of indices", "powers", "taxonomy"], "functions": {}, "parts": [{"gaps": [{"type": "jme", "variableReplacements": [], "vsetrangepoints": 5, "checkingtype": "absdiff", "showpreview": true, "showCorrectAnswer": true, "checkvariablenames": false, "answer": "a^{x+y}", "showFeedbackIcon": true, "marks": "2", "variableReplacementStrategy": "originalfirst", "scripts": {"mark": {"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}", "order": "after"}}, "vsetrange": [0, 1], "expectedvariablenames": [], "checkingaccuracy": 0.001, "musthave": {"showStrings": true, "partialCredit": 0, "strings": ["^"], "message": "

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

"}, "notallowed": {"showStrings": true, "partialCredit": 0, "strings": ["*", "(", ")", "+"], "message": "

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

"}}], "type": "gapfill", "stepsPenalty": "1", "marks": 0, "variableReplacementStrategy": "originalfirst", "prompt": "

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

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

", "scripts": {}, "showCorrectAnswer": true, "variableReplacements": [], "steps": [{"type": "information", "variableReplacements": [], "marks": 0, "variableReplacementStrategy": "originalfirst", "prompt": "

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

", "scripts": {}, "showCorrectAnswer": true, "showFeedbackIcon": true}], "showFeedbackIcon": true}, {"gaps": [{"type": "jme", "variableReplacements": [], "vsetrangepoints": 5, "checkingtype": "absdiff", "showpreview": true, "showCorrectAnswer": true, "checkvariablenames": false, "answer": "{c*d}*a^{p+q}", "showFeedbackIcon": true, "marks": "2", "variableReplacementStrategy": "originalfirst", "scripts": {"mark": {"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}", "order": "after"}}, "vsetrange": [0, 1], "expectedvariablenames": [], "checkingaccuracy": 0.001, "musthave": {"showStrings": true, "partialCredit": 0, "strings": ["^"], "message": "

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

"}, "notallowed": {"showStrings": true, "partialCredit": 0, "strings": ["*", "( )"], "message": "

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

"}}], "type": "gapfill", "variableReplacements": [], "marks": 0, "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]].

", "variableReplacementStrategy": "originalfirst", "scripts": {}, "showCorrectAnswer": true, "showFeedbackIcon": true}, {"gaps": [{"type": "jme", "variableReplacements": [], "vsetrangepoints": 5, "checkingtype": "absdiff", "showpreview": true, "showCorrectAnswer": true, "checkvariablenames": false, "answer": "{b/g}a^{x-y}", "showFeedbackIcon": true, "marks": "2", "variableReplacementStrategy": "originalfirst", "scripts": {"mark": {"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}", "order": "after"}}, "vsetrange": [0, 1], "expectedvariablenames": [], "checkingaccuracy": 0.001}], "type": "gapfill", "stepsPenalty": "1", "marks": 0, "variableReplacementStrategy": "originalfirst", "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.

"}, "notallowed": {"showStrings": true, "partialCredit": 0, "strings": ["*", "(", ")", ")"], "message": "

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

"}}], "type": "gapfill", "stepsPenalty": 0, "marks": 0, "variableReplacementStrategy": "originalfirst", "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.

"}, "notallowed": {"showStrings": true, "partialCredit": 0, "strings": ["sqrt", "root"], "message": "

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

"}}], "type": "gapfill", "stepsPenalty": "1", "marks": 0, "variableReplacementStrategy": "originalfirst", "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.

"}, "notallowed": {"showStrings": true, "partialCredit": 0, "strings": ["root"], "message": "

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

"}}], "type": "gapfill", "stepsPenalty": "1", "marks": 0, "variableReplacementStrategy": "originalfirst", "prompt": "

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

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

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

", "scripts": {}, "showCorrectAnswer": true, "showFeedbackIcon": true}], "showFeedbackIcon": true}], "variablesTest": {"condition": "", "maxRuns": 100}, "variables": {"g": {"templateType": "anything", "name": "g", "definition": "random(2..9 except b)", "description": "

Used in part c

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

Used in parts b,d and f

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

Used in parts b and e

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

Used in parts b and d

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

Used in parts a,c and e

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

Used in parts b,d and f

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

Used in parts a,c and f

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

Used in part c

", "group": "Ungrouped variables"}}, "variable_groups": [], "preamble": {"js": "", "css": ""}, "name": "Ed's copy of Laws of Indices", "extensions": [], "rulesets": {}, "metadata": {"description": "

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

", "licence": "Creative Commons Attribution 4.0 International"}, "ungrouped_variables": ["x", "y", "c", "d", "p", "q", "b", "g"], "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": "Ed Southwood", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2415/"}]}]}], "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": "Ed Southwood", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2415/"}]}