How do you write $\sqrt[5]{{(7 x)}^{4}}$ $root5((7x)^4)$ as an equivalent expression using a fractional exponent?

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How do you write $\sqrt[5]{{(7 x)}^{4}}$ $root5((7x)^4)$ as an equivalent expression using a fractional exponent?

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Answer 1 · 2017-01-10T17:17:59

${(7 x)}^{\frac{4}{5}}$ $(7x)^(4/5)$

Explanation:

First, we need to convert this radical form to an exponent form using this rule:

$\sqrt[n]{x} = x^{\frac{1}{n}}$ $root(color(red)(n))(color(blue)(x)) = color(blue)(x)^(1/color(red)(n))$

Using this rule:

$\sqrt[5]{{(7 x)}^{4}} \to {({(7 x)}^{4})}^{\frac{1}{5}}$ $root(color(red)(5))(color(blue)((7x)^4)) -> (color(blue)((7x)^4))^color(red)(1/5)$

We can now use this following rule for exponents to further simplify:

${(x^{a})}^{b} = x^{a \times b}$ $(x^color(red)(a))^color(blue)(b) = x^(color(red)(a)xxcolor(blue)(b))$

${({(7 x)}^{4})}^{\frac{1}{5}} \to {(7 x)}^{4 \times \frac{1}{5}} \to {(7 x)}^{\frac{4}{5}}$ $(color(blue)((7x)^4))^color(red)(1/5) -> color(blue)((7x))^(4xxcolor(red)(1/5)) -> (7x)^(4/5)$

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How do you write $\sqrt[5]{{(7 x)}^{4}}$ $root5((7x)^4)$ as an equivalent expression using a fractional exponent?

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