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The following questions show the method to use to solve the above questions:

Part a)

i) $[\\frac{3^2.2^4}{3^{-2}.2^5}]^2$

$[\\frac{3^2}{3^{-2}}]^2.[\\frac{2^4}{2^5}]^2$

$[3^4]^2.[2^{-1}]^2 = 3^6.2^{-2}$

ii) $[\\frac{5^3.7^2}{5^{2}.7^4}]^{-1}$

$[5^1]^{-1}.[7^{-2}]^{-1} = 5^{-1}.7^2$

iii) $\\frac{24x^2.y^{-4}.z^3}{8x^3.y^7.z^{-3}}$

$\\frac{24}{8}.x^{2-3}.y^{-4-7}.z^{3+3} = 3.x^{-1}.y^{-11}.z^{6}$

iv) $[\\frac{3a^2.b^{3}.c^{-2}}{2a^4.b.c^{2}}]^2$

$[\\frac{3}{2}]^2.[a^{2-4}]^2.[b^{3-1}]^2.[c^{-2-2}]^2 = \\frac{9}{4}.a^{-4}.b^{4}.c^{-8}$

Part b)

i) $\\frac{5x^3.y^{4}.z}{2x^5.y^2.z^{3}} \\times \\frac{3x^7.y^{2}.z^{-5}}{x^{-8}.y^5.z}$

$\\frac{5 \\times 3}{2}.x^{3-5+7+8}.y^{4-2+2-5}.z^{1-3-5-1} = \\frac{15}{2}.x^{13}.y^{-5}.z^{-8}$

ii) $\\frac{3a^5.b^{-2}.c^{7}}{2a.b^{-3}.c^{5}} \\div \\frac{a^4.b^{3}.c}{a^{-2}.b^6.c^{-1}}$

$\\frac{3}{2}.a^{5-1+4+2}.b^{-2+3+3-6}.c^{7-5+1+1} = \\frac{3}{2}.a^{10}.b^{-2}.c^{4}$

iii) $4^{\\frac{3}{2}}.8^{\\frac{2}{3}}.16^{\\frac{-3}{4}}$

$2^{\\frac{3 \\times2}{2}}.2^{\\frac{2 \\times3}{3}}.2^{\\frac{-3 \\times4}{4}}$

$2^{3+2-3} = 2^2$

iv) $[\\frac{49a^4.b^{-3}.c}{25a^{-2}.b^{3}.c^{5}}]^{\\frac{1}{2}}$

$[\\frac{49}{25}]^{\\frac{1}{2}}.[a^{4+2}]^{\\frac{1}{2}}.[b^{-3-3}]^{\\frac{1}{2}}.[c^{1-5}]^{\\frac{1}{2}} = \\frac{7}{5}.a^{3}.b^{-3}.c^{-2}$

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Simplify each of the following, without using a calculator:

i) $[\\frac{3^{\\var{num11[0]}}.9^{\\var{num11[1]}}}{3^{-\\var{num11[2]}}.27^{\\var{num11[3]}}}]^{\\var{num111}}$

${ans}^{power} =$ {[[0]]}^{[[1]]}

ii) $[\\frac{5^{\\var{num12[0]}}.7^{\\var{num12[1]}}}{5^{\\var{num12[2]}}.7^{\\var{num12[3]}}}]^{-\\var{num121}}$

${ans}^{power} =$ {[[2]]}^{[[3]]} {[[4]]}^{[[5]]}

iii) $\\frac{\\var{num132}x^{\\var{num13[0]}}.y^{-\\var{num13[1]}}.z^{\\var{num13[2]}}}{\\var{num131}x^{\\var{num13[3]}}.y^{\\var{num13[4]}}.z^{-\\var{num13[5]}}}$

${ans}^{power} =$ {[[6]]} x^{[[7]]} y^{[[8]]} z^{[[9]]}

iv) $(\\frac{\\var{num142}a^{\\var{num14[0]}}.b^{-\\var{num14[1]}}.c^{-\\var{num14[2]}}}{\\var{num141}a^{\\var{num14[3]}}.b.c^{\\var{num14[4]}}})^{\\var{num143}}$

${ans}^{power} =$ {[[10]]} a^{[[11]]} b^{[[12]]} c^{[[13]]}

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Simplify each of the following, without using a calculator:

i) $\\frac{\\var{num211[0]}x^{\\var{num21[0]}}.y^{\\var{num21[1]}}.z}{\\var{num211[1]}x^{\\var{num21[2]}}.y^{\\var{num21[3]}}.z^{\\var{num21[4]}}} \\times \\frac{\\var{num211[2]}x^{\\var{num21[5]}}.y^{\\var{num21[6]}}.z^{-\\var{num21[7]}}}{x^{-\\var{num21[8]}}.y^{\\var{num21[9]}}.z}$

{Fraction part}${ans}^{power}$ ={ [[0]]/[[1]]}x^{[[2]]} y^{[[3]]} z^{[[4]]}

ii) $\\frac{\\var{num221[0]}a^{\\var{num22[0]}}.b^{-\\var{num22[1]}}.c^{\\var{num22[2]}}}{\\var{num221[1]}a.b^{-\\var{num22[3]}}.c^{\\var{num22[4]}}} \\div \\frac{a^{-\\var{num22[7]}}.b^{\\var{num22[8]}}.c^{-\\var{num22[9]}}}{\\var{num221[2]}a^{\\var{num22[5]}}.b^{\\var{num22[6]}}.c}$

{Fraction part}${ans}^{power}$ = {[[5]]/[[6]]}a^{[[7]]} b^{[[8]]} c^{[[9]]}

iii) $4^{\\frac{\\var{num31[0]}}{2}}.8^{\\frac{\\var{num31[1]}}{3}}.16^{\\frac{-\\var{num31[2]}}{4}}$

${ans}^{power}$ = {[[10]]}^{[[11]]}

iv) $27^{\\frac{\\var{num32[0]}}{3}}.8^{\\frac{-\\var{num32[1]}}{3}}.16^{\\frac{\\var{num32[2]}}{4}}$

{ans}^{power} = {[[12]]}^{[[13]]}.{[[14]]}^{[[15]]}

v) $[\\frac{\\var{num321[0]}a^{\\var{num33[0]}}.b^{-\\var{num33[1]}}.c}{\\var{num321[1]}a^{-\\var{num33[2]}}.b^{\\var{num33[3]}}.c^{\\var{num33[4]}}}]^{\\frac{1}{2}}$

{Fraction part}${ans}^{power}$ = {[[16]]}/{[[17]]} a^{[[18]]}b^{[[19]]}c^{[[20]]}

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"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "showPrecisionHint": false, "type": "numberentry"}, {"integerPartialCredit": 0, "integerAnswer": true, "allowFractions": false, "variableReplacements": [], "maxValue": "{ans22b}", "minValue": "{ans22b}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "showPrecisionHint": false, "type": "numberentry"}, {"integerPartialCredit": 0, "integerAnswer": true, "allowFractions": false, "variableReplacements": [], "maxValue": "{ans22c}", "minValue": "{ans22c}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "showPrecisionHint": false, "type": "numberentry"}, {"integerPartialCredit": 0, "integerAnswer": true, "allowFractions": false, "variableReplacements": [], "maxValue": "{ans22d}", "minValue": "{ans22d}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "showPrecisionHint": false, "type": "numberentry"}, {"integerPartialCredit": 0, "integerAnswer": true, "allowFractions": false, "variableReplacements": [], "maxValue": "{ans22e}", "minValue": "{ans22e}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "showPrecisionHint": false, "type": "numberentry"}, {"allowFractions": false, "variableReplacements": [], "maxValue": "2", "minValue": "2", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "showPrecisionHint": false, "type": "numberentry"}, {"allowFractions": false, "variableReplacements": [], "maxValue": "{ans31}", "minValue": "{ans31}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "showPrecisionHint": false, "type": "numberentry"}, {"allowFractions": false, "variableReplacements": [], "maxValue": "3", "minValue": "3", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "showPrecisionHint": false, "type": "numberentry"}, {"allowFractions": false, "variableReplacements": [], "maxValue": "{num32[0]}", "minValue": "{num32[0]}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "showPrecisionHint": false, "type": "numberentry"}, {"allowFractions": false, "variableReplacements": [], "maxValue": "2", "minValue": "2", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "showPrecisionHint": false, "type": "numberentry"}, {"allowFractions": false, "variableReplacements": [], "maxValue": "{ans32}", "minValue": "{ans32}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "showPrecisionHint": false, "type": "numberentry"}, {"allowFractions": false, "variableReplacements": [], "maxValue": "{ans33a}", "minValue": "{ans33a}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "showPrecisionHint": false, "type": "numberentry"}, {"allowFractions": false, "variableReplacements": [], "maxValue": "{ans33b}", "minValue": "{ans33b}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "showPrecisionHint": false, "type": "numberentry"}, {"allowFractions": true, "variableReplacements": [], "maxValue": "{ans33c}", "minValue": "{ans33c}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "showPrecisionHint": false, "type": "numberentry"}, {"allowFractions": true, "variableReplacements": [], "maxValue": "{ans33d}", "minValue": "{ans33d}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "showPrecisionHint": false, "type": "numberentry"}, {"allowFractions": true, "variableReplacements": [], "maxValue": "{ans33e}", "minValue": "{ans33e}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "showPrecisionHint": false, "type": "numberentry"}], "steps": [{"prompt": "

Break each individual number down to a simple power first and then combine together.

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Simplify the Indices

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variables", "name": "num121", "description": ""}, "num32": {"definition": "shuffle(2..8)[0..3]", "templateType": "anything", "group": "Ungrouped variables", "name": "num32", "description": ""}, "ans22d": {"definition": "-num22[1]+num22[3]+num22[6]-num22[8]", "templateType": "anything", "group": "Ungrouped variables", "name": "ans22d", "description": ""}, "ans22e": {"definition": "num22[2]-num22[4]+1+num22[9]", "templateType": "anything", "group": "Ungrouped variables", "name": "ans22e", "description": ""}, "ans22b": {"definition": "num221[1]", "templateType": "anything", "group": "Ungrouped variables", "name": "ans22b", "description": ""}, "ans22c": {"definition": "num22[0]-1+num22[5]+num22[7]", "templateType": "anything", "group": "Ungrouped variables", "name": "ans22c", "description": ""}, "ans22a": {"definition": "num221[0]*num221[2]", "templateType": "anything", "group": "Ungrouped variables", "name": "ans22a", "description": ""}, "num22": {"definition": "shuffle(2..11)[0..10]", "templateType": "anything", "group": "Ungrouped variables", "name": "num22", "description": ""}, "num21": {"definition": "shuffle(2..11)[0..10]", "templateType": "anything", "group": "Ungrouped variables", "name": "num21", "description": ""}, "num12": {"definition": "shuffle(2..5)[0..4]", "templateType": "anything", "group": "Ungrouped variables", "name": "num12", "description": ""}, "ans14a1": {"definition": "random(2..4)", "templateType": "anything", "group": "Ungrouped variables", "name": "ans14a1", "description": ""}, "num13": {"definition": "shuffle(2..9)[0..6]", "templateType": "anything", "group": "Ungrouped variables", "name": "num13", "description": ""}, "num211": {"definition": "shuffle(3..7 except[4,6])[0..3]", "templateType": "anything", "group": "Ungrouped variables", "name": "num211", "description": ""}, "ans11": {"definition": "(num11[0]+(2*num11[1])+num11[2]-(3*num11[3]))*num111", "templateType": "anything", "group": "Ungrouped variables", "name": "ans11", "description": ""}, "ans12a": {"definition": "(num12[0]-num12[2])*(-num121)", "templateType": "anything", "group": "Ungrouped variables", "name": "ans12a", "description": ""}, "ans31": {"definition": "num31[0]+num31[1]-num31[2]", "templateType": "anything", "group": "Ungrouped variables", "name": "ans31", "description": ""}, "ans32": {"definition": "-num31[1]+num31[2]", "templateType": "anything", "group": "Ungrouped variables", "name": "ans32", "description": ""}, "ans14aT": {"definition": "ans14a1^num143", "templateType": "anything", "group": "Ungrouped variables", "name": "ans14aT", "description": ""}, "ans14c": {"definition": "(-num14[1]-1)*num143", "templateType": "anything", "group": "Ungrouped variables", "name": "ans14c", "description": ""}, "ans14b": {"definition": "(num14[0]-num14[3])*num143", "templateType": "anything", "group": "Ungrouped variables", "name": "ans14b", "description": ""}, "ans14a": {"definition": "(num142/num141)^num143", "templateType": "anything", "group": "Ungrouped variables", "name": "ans14a", "description": ""}, "num111": {"definition": "random(2..3)", "templateType": "anything", "group": "Ungrouped variables", "name": "num111", "description": ""}, "num132": {"definition": "num131*ans13a", "templateType": "anything", "group": "Ungrouped variables", "name": "num132", "description": ""}, "num131": {"definition": "random(2,4,8,5)", "templateType": "anything", "group": "Ungrouped variables", "name": "num131", "description": ""}, "num31": {"definition": "shuffle(2..8)[0..3]", "templateType": "anything", "group": "Ungrouped variables", "name": "num31", "description": ""}, "ans21e": {"definition": "1-num21[4]-num21[7]-1", "templateType": "anything", "group": "Ungrouped variables", "name": "ans21e", "description": ""}, "ans21d": {"definition": "num21[1]-num21[3]+num21[6]-num21[9]", "templateType": "anything", "group": "Ungrouped variables", "name": "ans21d", "description": ""}, "ans21c": {"definition": "num21[0]-num21[2]+num21[5]+num21[8]", "templateType": "anything", "group": "Ungrouped variables", "name": "ans21c", "description": ""}, "ans21b": {"definition": "num211[1]", "templateType": "anything", "group": "Ungrouped variables", "name": "ans21b", "description": ""}, "ans21a": {"definition": "num211[0]*num211[2]", "templateType": "anything", "group": "Ungrouped variables", "name": "ans21a", "description": ""}, "num11": {"definition": "shuffle(2..5)[0..4]", "templateType": "anything", "group": "Ungrouped variables", "name": "num11", "description": ""}, "ans33d": {"definition": "(-num33[1]-num33[3])/2", "templateType": "anything", "group": "Ungrouped variables", "name": "ans33d", "description": ""}, "ans33e": {"definition": "(1-num33[4])/2", "templateType": "anything", "group": "Ungrouped variables", "name": "ans33e", "description": ""}, "num33": {"definition": "shuffle(2..8)[0..5]", "templateType": "anything", "group": "Ungrouped variables", "name": "num33", "description": ""}, "ans33a": {"definition": "sqrt(num321[0])", "templateType": "anything", "group": "Ungrouped variables", "name": "ans33a", "description": ""}, "ans33b": {"definition": "sqrt(num321[1])", "templateType": "anything", "group": "Ungrouped variables", "name": "ans33b", "description": ""}, "ans33c": {"definition": "(num33[0]+num33[2])/2", "templateType": "anything", "group": "Ungrouped variables", "name": "ans33c", "description": ""}}, "metadata": {"description": "

Indices in their simplest form

rebelmaths

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