Used in part c
"}, "g": {"group": "Ungrouped variables", "definition": "random(2..9 except b)", "templateType": "anything", "name": "g", "description": "Used in part c
"}, "y": {"group": "Ungrouped variables", "definition": "random(2..6)", "templateType": "anything", "name": "y", "description": "\nUsed in parts a,c and f
"}, "p": {"group": "Ungrouped variables", "definition": "random(2..9)", "templateType": "anything", "name": "p", "description": "Used in parts b and d
"}, "x": {"group": "Ungrouped variables", "definition": "random(2..9 except y)", "templateType": "anything", "name": "x", "description": "Used in parts a,c and e
"}, "q": {"group": "Ungrouped variables", "definition": "random(3,5)", "templateType": "anything", "name": "q", "description": "Used in parts b,d and f
"}, "d": {"group": "Ungrouped variables", "definition": "random(3..5)", "templateType": "anything", "name": "d", "description": "Used in parts b and e
"}, "c": {"group": "Ungrouped variables", "definition": "random(2,4)", "templateType": "anything", "name": "c", "description": "Used in parts b,d and f
"}}, "functions": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "metadata": {"description": "This question aims to test understanding and ability to use the laws of indices.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "Using the laws of indices, simplify each expression down to its simplest form. Recall that $a^{0} = 1$ for any number $a$.
", "advice": "Here we are using the rule of indices: $a^m \\times a^n = a^{m+n}$.
\nUsing this rule,
\n\\[
\\begin{align}
a^\\var{x} \\times a^\\var{y}\\ &= a^\\simplify[all, !collectNumbers]{{x}+{y}}\\\\
&= a^\\var{x+y}.
\\end{align}
\\]
We are asked to find $\\var{c}a^\\var{p} \\times \\var{d}a^\\var{q}$.
\nNotice there is a constant in front of each of the terms.
\nTo 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}.\\]
\nWe know that $\\var{c} \\times \\var{d} = \\var{c*d}$, and using the rule of indices: $a^\\var{p} \\times a^\\var{q} = a^\\var{p+q}$.
\nTherefore:
\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}
Here we are using: $a^m \\div a^n = a^{m-n}$.
\nWe are asked to simplify the expression, $\\displaystyle\\simplify{{b}*a^{x}/({g}*a^{y})}$.
\nTo 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})}.\\]
\nUsing 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}
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}
Alternatively,
\nUsing 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}
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}
The question asks us to simplify $(\\simplify{{c}*a^{p}})^{\\var{q}}$.
\nTo do this we use the rules:
\n\\[(a^{m})^{n} = a^{mn},\\]
\n\\[(ab)^m = a^mb^m.\\]
\nWe can then expand the equation as
\n\\[(\\simplify{{c}*a^{p}})^{\\var{q}}= \\var{c}^{\\var{q}} \\times (a^{\\var{p}})^{\\var{q}}.\\]
\nThen 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}
\\]
The question asks us to simplify $\\sqrt[\\var{d}]{\\var{x}^\\var{d}a}$.
\nTo do this we use the rules:
\n\\[a^\\frac{1}{m} = \\sqrt[m]{a},\\]
\n\\[(ab)^m = a^mb^m.\\]
\nWe 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}
\\]
The question requires us to simplify $\\sqrt[\\var{c}]{a^\\var{q}}$.
\nHere, 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}
\\]
You must write your answer as a single power of a.
", "strings": ["*", "(", ")", "+"], "partialCredit": 0}, "musthave": {"showStrings": true, "message": "You must write your answer as a single power of a.
", "strings": ["^"], "partialCredit": 0}, "checkingtype": "absdiff", "answer": "a^{x+y}", "showCorrectAnswer": true, "checkingaccuracy": 0.001, "marks": "2", "vsetrange": [0, 1], "vsetrangepoints": 5, "type": "jme", "checkvariablenames": false, "variableReplacements": []}], "type": "gapfill", "steps": [{"scripts": {}, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "marks": 0, "showFeedbackIcon": true, "type": "information", "prompt": "Use the rule: $a^m \\times a^n = a^{m+n}$.
", "variableReplacements": []}], "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]].
", "variableReplacements": []}, {"scripts": {}, "gaps": [{"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"}}, "showpreview": true, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "expectedvariablenames": [], "notallowed": {"showStrings": true, "message": "You must write your answer as a number multiplied by a single power of a.
", "strings": ["*", "( )"], "partialCredit": 0}, "musthave": {"showStrings": true, "message": "You must write your answer as a number multiplied by a single power of a.
", "strings": ["^"], "partialCredit": 0}, "checkingtype": "absdiff", "answer": "{c*d}*a^{p+q}", "showCorrectAnswer": true, "checkingaccuracy": 0.001, "marks": "2", "vsetrange": [0, 1], "vsetrangepoints": 5, "type": "jme", "checkvariablenames": false, "variableReplacements": []}], "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "marks": 0, "showFeedbackIcon": true, "type": "gapfill", "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", "variableReplacements": []}, {"scripts": {}, "stepsPenalty": "1", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "marks": 0, "showFeedbackIcon": true, "gaps": [{"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"}}, "showpreview": true, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "expectedvariablenames": [], "checkingtype": "absdiff", "answer": "{b/g}a^{x-y}", "showCorrectAnswer": true, "checkingaccuracy": 0.001, "marks": "2", "vsetrange": [0, 1], "vsetrangepoints": 5, "type": "jme", "checkvariablenames": false, "variableReplacements": []}], "type": "gapfill", "steps": [{"scripts": {}, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "marks": 0, "showFeedbackIcon": true, "type": "information", "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": []}], "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", "variableReplacements": []}, {"scripts": {}, "stepsPenalty": 0, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "marks": 0, "showFeedbackIcon": true, "gaps": [{"scripts": {}, "showpreview": true, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "expectedvariablenames": [], "notallowed": {"showStrings": true, "message": "You must enter your answer as an integer multiplied by a single power of a.
", "strings": ["*", "(", ")", ")"], "partialCredit": 0}, "musthave": {"showStrings": true, "message": "You must enter your answer as an integer multiplied by a single power of a.
", "strings": ["^"], "partialCredit": 0}, "checkingtype": "absdiff", "answer": "{c^{q}}*a^{p*q}", "showCorrectAnswer": true, "checkingaccuracy": 0.001, "marks": "2", "vsetrange": [0, 1], "vsetrangepoints": 5, "type": "jme", "checkvariablenames": false, "variableReplacements": []}], "type": "gapfill", "steps": [{"scripts": {}, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "marks": 0, "showFeedbackIcon": true, "type": "information", "prompt": "Use the rules:
\n$(ab)^m = a^mb^m$.
\n$(a^m)^n = a^{mn}$.
", "variableReplacements": []}], "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", "variableReplacements": []}, {"scripts": {}, "stepsPenalty": "1", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "marks": 0, "showFeedbackIcon": true, "gaps": [{"scripts": {}, "showpreview": true, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "expectedvariablenames": [], "answersimplification": "basic", "notallowed": {"showStrings": true, "message": "You must input your answer as a single power of a.
", "strings": ["sqrt", "root"], "partialCredit": 0}, "musthave": {"showStrings": true, "message": "You must input your answer as a single power of a.
", "strings": ["^", "(", ")"], "partialCredit": 0}, "checkingtype": "absdiff", "answer": "a^(1/{d})", "showCorrectAnswer": true, "checkingaccuracy": 0.001, "marks": "2", "vsetrange": [0, 1], "vsetrangepoints": 5, "type": "jme", "checkvariablenames": false, "variableReplacements": []}], "type": "gapfill", "steps": [{"scripts": {}, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "marks": 0, "showFeedbackIcon": true, "type": "information", "prompt": "Use the rule: $a^\\frac{1}{m} = \\sqrt[m]{a}$.
", "variableReplacements": []}], "prompt": "\nWrite $\\sqrt[\\var{d}]{a}$ as a single power of $a$.
\n$\\sqrt[\\var{d}]{a} =$ [[0]].
\n\n", "variableReplacements": []}, {"scripts": {}, "stepsPenalty": "1", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "marks": 0, "showFeedbackIcon": true, "gaps": [{"scripts": {}, "showpreview": true, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "expectedvariablenames": [], "notallowed": {"showStrings": true, "message": "You must enter your answer as a single power of a.
", "strings": ["root"], "partialCredit": 0}, "musthave": {"showStrings": true, "message": "You must enter your answer as a single power of a.
", "strings": ["^", "(", ")"], "partialCredit": 0}, "checkingtype": "absdiff", "answer": "a^({c}/{q})", "showCorrectAnswer": true, "checkingaccuracy": 0.001, "marks": "2", "vsetrange": [0, 1], "vsetrangepoints": 5, "type": "jme", "checkvariablenames": false, "variableReplacements": []}], "type": "gapfill", "steps": [{"scripts": {}, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "marks": 0, "showFeedbackIcon": true, "type": "information", "prompt": "Use the rule: $a^\\frac{n}{m} = \\sqrt[m]{a^n}$.
", "variableReplacements": []}], "prompt": "Write $\\sqrt[\\var{q}]{a^\\var{c}}$ as a single power of $a$.
\n$\\sqrt[\\var{q}]{a^\\var{c}} =$ [[0]].
\n", "variableReplacements": []}], "preamble": {"js": "", "css": ""}, "tags": ["indices", "laws of indices", "powers", "taxonomy"], "ungrouped_variables": ["x", "y", "c", "d", "p", "q", "b", "g"], "rulesets": {}, "variable_groups": [], "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": "David Goulding", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2365/"}]}]}], "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": "David Goulding", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2365/"}]}