// Numbas version: finer_feedback_settings {"name": "Indices: fractional powers (non-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 (non-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 $(\\sqrt{\\var{base1}})^2=\\var{base1}$.

By using index laws you should see that $(\\var{base1}^{1/2})^2=\\var{base1}$.

The above equations imply that $\\sqrt{\\var{base1}}$ 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{\\var{base1}})^2=\\var{base1}=(\\var{base1}^{1/2})^2\\]

we can say \\[\\sqrt{\\var{base1}}=\\var{base1}^{1/2}\\]

Which we would type in as $\\var{base1}\\wedge(1/2)$.

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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 $(\\sqrt[3]{\\var{base2}})^3=\\var{base2}$.

By using index laws you should see that $(\\var{base2}^{1/3})^3=\\var{base2}$.

The above equations imply that $\\sqrt[3]{\\var{base2}}$ 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]{\\var{base2}})^3=\\var{base2}=(\\var{base2}^{1/3})^3\\]

we can say \\[\\sqrt[3]{\\var{base2}}=\\var{base2}^{1/3}\\]

Which we would type in as $\\var{base2}\\wedge(1/3)$.

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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.

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

$\\sqrt[\\var{root1}]{\\var{base3}}$ = [[0]]

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

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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.

$\\displaystyle\\left(\\sqrt[\\var{root2}]{\\var{base4}}\\right)^\\var{power2}$ = [[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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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.

$\\sqrt[\\var{root3}]{\\var{base1}^\\var{power3}}$ = [[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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Use ^ for powers. Input your answer in index form.

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

By interpreting the denominator of the fractional power as an n^th root, determine the value of the following:

$\\var{square[0]}^{\\frac{1}{2}}$ = [[0]]

$\\var{cube[0]}^{\\frac{2}{3}}$ = [[1]]

$\\var{onezeros}^{\\frac{1}{\\var{zeros}}}$ = [[2]]

Convert the denominator of the fractional power to a root.

For example, $27^{\\frac{2}{3}}=\\sqrt[3]{27}^2=3^2=9$.

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