// Numbas version: exam_results_page_options {"name": "Indices: fractional powers (algebraic)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Indices: fractional powers (algebraic)", "tags": ["exponent", "exponents", "fractional", "index", "index laws", "Index Laws", "indices", "power", "powers", "rational", "roots"], "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "

Simplify the following without the use of a calculator. Write your answer in index form using ^ to signify powers.

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By using the definition of the square root you should see that $(\\sqrt{q})^2=q$.

By using index laws you should see that $(q^{1/2})^2=q$.

The above equations imply that $\\sqrt{q}$ can also be written as [[0]].

Note: If you want to use a fraction as a power you should use brackets to surround your power, for example, type 12^(2/3) for $12^\\frac{2}{3}$.

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Given \\[(\\sqrt{q})^2=q=(q^{1/2})^2\\]

we can say \\[\\sqrt{q}=q^{1/2}\\]

Which we would type in as $q\\wedge(1/2)$.

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Your answer is longer than necessary.

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Use ^ for powers. Input your answer in index form.

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Use ^ for powers. Input your answer in index form.

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By using the definition of the cube root you should see that $(\\sqrt[3]{p})^3=p$.

By using index laws you should see that $(p^{1/3})^3=p$.

The above equations imply that $\\sqrt[3]{p}$ can also be written as [[0]].

Note: If you want to use a fraction as a power you should use brackets to surround your power, for example, type 12^(2/3) for $12^\\frac{2}{3}$.

Given \\[(\\sqrt[3]{p})^3=p=(p^{1/3})^3\\]

we can say \\[\\sqrt[3]{p}=p^{1/3}\\]

Which we would type in as $p\\wedge(1/3)$.

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Use ^ for powers. Input your answer in index form.

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Use ^ for powers. Input your answer in index form.

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Use the same approach you used in the above questions to simplify the following in index form.

$\\sqrt[\\var{root1}]{g}$ = [[0]]

By the same reasoning as used in the above questions we have $\\sqrt[n]{a}=a^{\\frac{1}{n}}$.

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Use ^ for powers. Input your answer in index form.

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$\\displaystyle\\left(\\sqrt[\\var{root2}]{e}\\right)^\\var{power2}$ = [[0]]

Convert the root to a fractional power and then use the index laws to deal with the two different powers.

For example, \\[\\sqrt[3]{2}^5=(2^{\\frac{1}{3}})^5=2^{\\frac{5}{3}}\\]

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$\\sqrt[\\var{root3}]{r^\\var{power3}}$ = [[0]]

Convert the root to a fractional power and then use the index laws to deal with the two different powers.

For example, \\[\\sqrt[3]{2^5}=(2^5)^{\\frac{1}{3}}=2^{\\frac{5}{3}}\\]

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