// Numbas version: exam_results_page_options {"name": "Algebra IV: Properties of indices (2) - Fractions", "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": "Algebra IV: Properties of indices (2) - Fractions", "tags": [], "advice": "
Recall the laws of indices:
\n$x^a \\times x^b = x^{a+b}$
$x^a \\div x^b = x^{a-b}$
$x^{-a} = \\frac{1}{x^a}$
$(x^a)^b = x^{ab}$
$(\\frac{x}{y})^a = \\frac{x^a}{y^a}$
$x^\\frac{a}{b} = (\\sqrt[b]{x})^{a}$
$x^0 = 1$
$\\var{c}^{\\frac{1}{\\var{b}}}$
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", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "marks": 1, "scripts": {}, "showCorrectAnswer": true, "type": "numberentry", "showPrecisionHint": false}, {"stepsPenalty": 0, "prompt": "$\\var{d}^{\\frac{1}{\\var{a}}}$
", "allowFractions": false, "variableReplacements": [], "maxValue": "3", "minValue": "3", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "steps": [{"prompt": "Hint: $x^{1/z}$ is the same as $\\sqrt[z]{x}$
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "marks": 1, "scripts": {}, "showCorrectAnswer": true, "type": "numberentry", "showPrecisionHint": false}, {"stepsPenalty": 0, "prompt": "$\\var{h}^{\\frac{\\var{f}}{\\var{g}}}$
", "allowFractions": false, "variableReplacements": [], "maxValue": "3^{f}", "minValue": "3^{f}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "steps": [{"prompt": "Hint: $x^{y/z}$ can also be considered as $(x^{1/z})^{y}$.
\nAs you know, $x^{1/z} = \\sqrt[z]{x}$ therefore an alternative way of looking at the expression is $(\\sqrt[z]{x})^{y}$.
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", "allowFractions": false, "variableReplacements": [], "maxValue": "5^f1", "minValue": "5^f1", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "steps": [{"prompt": "Hint: $x^{y/z}$ can also be considered as $(x^{1/z})^{y}$.
\nAs you know, $x^{1/z} = \\sqrt[z]{x}$ therefore an alternative way of looking at the expression is $(\\sqrt[z]{x})^{y}$.
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", "allowFractions": false, "variableReplacements": [], "maxValue": "343", "minValue": "343", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "steps": [{"prompt": "Hint: $x^{y/z}$ can also be considered as $(x^{1/z})^{y}$.
\nAs you know, $x^{1/z} = \\sqrt[z]{x}$ therefore an alternative way of looking at the expression is $(\\sqrt[z]{x})^{y}$.
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\nHint: $x^{y/z}$ can also be considered as $(x^{1/z})^{y}$.
\nAs you know, $x^{1/z} = \\sqrt[z]{x}$ therefore an alternative way of looking at the expression is $(\\sqrt[z]{x})^{y}$.
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "marks": 1, "scripts": {}, "showCorrectAnswer": true, "type": "numberentry", "showPrecisionHint": false}, {"stepsPenalty": 0, "prompt": "$(\\var{t1}/\\var{t2})^{\\simplify{{f1}/{f2}}}$
", "allowFractions": true, "variableReplacements": [], "maxValue": "(2/3)^{f1}", "minValue": "(2/3)^{f1}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": true, "steps": [{"prompt": "Make sure to consider the brackets in this question.
\nHint: $x^{y/z}$ can also be considered as $(x^{1/z})^{y}$.
\nAs you know, $x^{1/z} = \\sqrt[z]{x}$ therefore an alternative way of looking at the expression is $(\\sqrt[z]{x})^{y}$.
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "marks": 1, "scripts": {}, "showCorrectAnswer": true, "type": "numberentry", "showPrecisionHint": false}], "extensions": [], "statement": "Simplify the following expressions, giving your answer in its simplest form.
\nGive your answer as either a fraction or an integer.
\nTry doing these questions without a calculator if you can to practise your use of powers and roots.
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