Q. 98 - Intermediate Algebra

Question

Use the Quotient Property to simplify square roots.

$\sqrt{\frac{125n^7}{169}}$ 

Step-by-Step Solution

Answer

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

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, $b\ne 0$for any integer $n\ge 2$.

Step 3. Calculation.

Simplifying the expression.

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