Hint: $x^{1/z}$ is the same as $\\sqrt[z]{x}$
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", "scripts": {}, "marks": 0}], "showPrecisionHint": false, "prompt": "$\\var{d}^{\\frac{1}{\\var{a}}}$
", "maxValue": "3"}, {"allowFractions": false, "variableReplacementStrategy": "originalfirst", "stepsPenalty": 0, "type": "numberentry", "correctAnswerFraction": false, "minValue": "3^{f}", "scripts": {}, "marks": 1, "showCorrectAnswer": true, "variableReplacements": [], "steps": [{"showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "type": "information", "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 $(\\sqrt[z]{x})^{y}$.
", "scripts": {}, "marks": 0}], "showPrecisionHint": false, "prompt": "$\\var{h}^{\\frac{\\var{f}}{\\var{g}}}$
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\nAs you know, $x^{1/z} = \\sqrt[z]{x}$ therefore an alternative way of looking at the expression is $(\\sqrt[z]{x})^{y}$.
", "scripts": {}, "marks": 0}], "showPrecisionHint": false, "prompt": "$\\var{f3}^{\\frac{\\var{f1}}{\\var{f2}}}$
", "maxValue": "5^f1"}, {"allowFractions": false, "variableReplacementStrategy": "originalfirst", "stepsPenalty": 0, "type": "numberentry", "correctAnswerFraction": false, "minValue": "343", "scripts": {}, "marks": 1, "showCorrectAnswer": true, "variableReplacements": [], "steps": [{"showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "type": "information", "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 $(\\sqrt[z]{x})^{y}$.
", "scripts": {}, "marks": 0}], "showPrecisionHint": false, "prompt": "$49^{\\frac{3}{2}}$
", "maxValue": "343"}, {"allowFractions": true, "variableReplacementStrategy": "originalfirst", "stepsPenalty": 0, "type": "numberentry", "correctAnswerFraction": true, "minValue": "5^-{f2}", "scripts": {}, "marks": 1, "showCorrectAnswer": true, "variableReplacements": [], "steps": [{"showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "type": "information", "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 $(\\sqrt[z]{x})^{y}$.
", "scripts": {}, "marks": 0}], "showPrecisionHint": false, "prompt": "$(1/\\var{f4})^{\\simplify{{f2}/{f1}}}$
", "maxValue": "5^-{f2}"}, {"allowFractions": true, "variableReplacementStrategy": "originalfirst", "stepsPenalty": 0, "type": "numberentry", "correctAnswerFraction": true, "minValue": "(2/3)^{f1}", "scripts": {}, "marks": 1, "showCorrectAnswer": true, "variableReplacements": [], "steps": [{"showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "type": "information", "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 $(\\sqrt[z]{x})^{y}$.
", "scripts": {}, "marks": 0}], "showPrecisionHint": false, "prompt": "$(\\var{t1}/\\var{t2})^{\\simplify{{f1}/{f2}}}$
", "maxValue": "(2/3)^{f1}"}], "question_groups": [{"pickQuestions": 0, "name": "", "pickingStrategy": "all-ordered", "questions": []}], "rulesets": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "functions": {}, "extensions": [], "variables": {"t1": {"definition": "2^f2", "name": "t1", "group": "Ungrouped variables", "description": "", "templateType": "anything"}, "f3": {"definition": "5^f2", "name": "f3", "group": "Ungrouped variables", "description": "", "templateType": "anything"}, "a": {"definition": "random(2..5)", "name": "a", "group": "Ungrouped variables", "description": "", "templateType": "anything"}, "k": {"definition": "random(2..5 except j)", "name": "k", "group": "Ungrouped variables", "description": "", "templateType": "anything"}, "t2": {"definition": "3^f2", "name": "t2", "group": "Ungrouped variables", "description": "", "templateType": "anything"}, "j": {"definition": "random(2..5)", "name": "j", "group": "Ungrouped variables", "description": "", "templateType": "anything"}, "f4": {"definition": "5^f1", "name": "f4", "group": "Ungrouped variables", "description": "", "templateType": "anything"}, "g": {"definition": "random(3..6)", "name": "g", "group": "Ungrouped variables", "description": "", "templateType": "anything"}, "f2": {"definition": "random(3..6)", "name": "f2", "group": "Ungrouped variables", "description": "", "templateType": "anything"}, "c": {"definition": "2^b", "name": "c", "group": "Ungrouped variables", "description": "", "templateType": "anything"}, "f1": {"definition": "random(2..f2-1)", "name": "f1", "group": "Ungrouped variables", "description": "", "templateType": "anything"}, "b": {"definition": "random(2..4)", "name": "b", "group": "Ungrouped variables", "description": "", "templateType": "anything"}, "f": {"definition": "random(2..g-1)", "name": "f", "group": "Ungrouped variables", "description": "", "templateType": "anything"}, "h": {"definition": "3^g", "name": "h", "group": "Ungrouped variables", "description": "", "templateType": "anything"}, "l": {"definition": "2^j", "name": "l", "group": "Ungrouped variables", "description": "", "templateType": "anything"}, "d": {"definition": "3^a", "name": "d", "group": "Ungrouped variables", "description": "", "templateType": "anything"}}, "ungrouped_variables": ["a", "b", "c", "d", "f", "g", "h", "j", "k", "l", "f1", "f2", "f3", "f4", "t1", "t2"], "name": "Simon's copy of Algebra IV: Properties of indices (2) - Fractions", "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.
", "tags": [], "showQuestionGroupNames": false, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "Simplifying indices.
"}, "advice": "Recall the laws of indices:
\n$x^a \\times x^b = x^{a+b}$
$x^a \\div x^b = x^{a-b}$
$x^{-a} = \\frac{1}{x^a}$
$(x^a)^b = x^{ab}$
$(\\frac{x}{y})^a = \\frac{x^a}{y^a}$
$x^\\frac{a}{b} = (\\sqrt[b]{x})^{a}$
$x^0 = 1$