Question

Use the Quotient Property to simplify square roots.

$\sqrt{\frac{125n^7}{169}}$

Short answer

The simplified form is$\frac{5\left|n^5\right|\sqrt{5n}}{13}$.

Step-by-step solution

Step 1. Given information.

The radical expression as:

$\sqrt{\frac{125n^7}{169}}$.

Step 2. Write the concept.

To determine a simplified form of a radical expression following formula is used:

$$\begin{gathered} \sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}and \\ \frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}} \end{gathered}$$

If $\sqrt[n]{a}and\sqrt[n]{b}$are real numbers, localid="1644695115626" $b\ne 0$for any integer $n\ge 2$.

Step 3. Calculation.

Simplifying the expression.

$$\begin{gathered} \sqrt{\frac{125n^7}{169}}=\sqrt{\frac{25n^6.5n}{169}} \\ =\frac{\sqrt{25n^6.5n}}{\sqrt{169}} \\ =\frac{\sqrt{{\left(5n^5\right)}^2}.\sqrt{5n}}{13} \\ =\frac{5\left|n^5\right|\sqrt{5n}}{13} \end{gathered}$$