// 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", "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]]

In general, we have $\\sqrt[n]{a}=a^{\\frac{1}{n}}$.

In particular, $\\sqrt[\\var{root1}]{\\var{base3}}=\\var{base3}^\\frac{1}{\\var{root1}}$.

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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.

$\\displaystyle\\left(\\sqrt[\\var{root2}]{\\var{base4}}\\right)^\\var{power2}$ = [[0]]

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

That is, \\begin{align}\\left(\\sqrt[\\var{root2}]{\\var{base4}}\\right)^\\var{power2}
&=\\left(\\var{base4}^{\\frac{1}{\\var{root2}}}\\right)^\\var{power2}\\\\
&=\\var{base4}^{\\frac{1}{\\var{root2}}\\times\\var{power2}}\\\\
&=\\var{base4}^{\\var[fractionnumbers]{power2/root2}}.\\end{align}

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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.

$\\sqrt[\\var{root3}]{\\var{base1}^\\var{power3}}$ = [[0]]

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

That is, \\begin{align}\\left(\\sqrt[\\var{root3}]{\\var{base1}}\\right)^\\var{power3}
&=\\left(\\var{base1}^{\\frac{1}{\\var{root3}}}\\right)^\\var{power3}\\\\
&=\\var{base1}^{\\frac{1}{\\var{root3}}\\times\\var{power3}}\\\\
&=\\var{base1}^{\\var[fractionnumbers]{power3/root3}}.\\end{align}

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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.

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 and use your multiplication knowledge.

That is,

\\begin{align}\\var{square[0]}^{\\frac{1}{2}}
&=\\sqrt{\\var{square[0]}}\\\\
&=\\var{square[1]}&(\\text{since } \\var{square[1]}^2=\\var{square[0]})\\end{align}

and
\\begin{align}\\var{cube[0]}^{\\frac{2}{3}}
&=\\left(\\var{cube[0]}^{\\frac{1}{3}}\\right)^2\\\\
&=\\left(\\sqrt[3]{\\var{cube[0]}}\\right)^2\\\\
&={\\var{sqrt(cube[1])}}^2&(\\text{since } \\var{sqrt(cube[1])}^3=\\var{cube[0]})\\\\
&=\\var{cube[1]},\\end{align}

and finally

\\begin{align}\\var{onezeros}^{\\frac{1}{\\var{zeros}}}
&=\\sqrt[\\var{zeros}]{\\var{onezeros}}\\\\
&=10&(\\text{since } 10^\\var{zeros}=\\var{onezeros}).\\end{align}

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