# How do you simplify root5(x^3)/root7(x^4)?

Jul 30, 2016

$\text{ "x^(3/5)/x^(4/7) = x^(1/35) = root(35)(x) larr" 35th root}$

#### Explanation:

Write as: (x^(3/5))/(x^(4/7)

This is the same as: ${x}^{\frac{3}{5} - \frac{4}{7}}$

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Consider the $\frac{3}{5} - \frac{4}{7}$

Write as $\left(\frac{3}{5} \times 1\right) - \left(\frac{4}{7} \times 1\right)$

This is the same as:$\left(\frac{3}{5} \times \frac{7}{7}\right) - \left(\frac{4}{7} \times \frac{5}{5}\right)$

$= \frac{21}{35} - \frac{20}{35} = \frac{21 - 20}{35} = \frac{1}{35}$
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$\textcolor{w h i t e}{.}$

So $\text{ "x^(3/5)/x^(4/7) = x^(1/35) = root(35)(x) larr" 35th root}$

Jul 30, 2016

${x}^{\frac{1}{20}} = \sqrt[20]{x}$

#### Explanation:

If both roots were the same we could have combined them into the root of a single fraction. But they are different.

Change to index form using : $\text{ } \sqrt[q]{{x}^{p}} = {x}^{\frac{p}{q}}$

$\frac{\sqrt[5]{{x}^{3}}}{\sqrt[7]{{x}^{4}}} = {x}^{\frac{3}{5}} / {x}^{\frac{4}{7}} \text{ simplify using} {x}^{m} / {x}^{n} = {x}^{m - n}$

=${x}^{\frac{3}{5} - \frac{4}{7}}$

=${x}^{\frac{21 - 20}{35}}$

=${x}^{\frac{1}{35}} = \sqrt[35]{x}$