Question

what is the simplified form of $sqrt{\frac{72x^{16}}{50x^{36}}}$? assume $x \
eq 0$.$\bigcirc$ $\frac{6}{5x^{10}}$$\bigcirc$ $\frac{6}{5x^{9}}$$\bigcirc$ $\frac{6}{5}x^{10}$$\bigcirc$ $\frac{6}{5}x^{9}$

Explanation:

Step1: Simplify the rational expression inside the square root

First, simplify the coefficients and the variable terms separately:
$$\frac{72x^{16}}{50x^{36}} = \frac{72}{50} \cdot x^{16-36} = \frac{36}{25} \cdot x^{-20}$$

Step2: Rewrite negative exponent as positive

Express $x^{-20}$ as $\frac{1}{x^{20}}$:
$$\frac{36}{25} \cdot x^{-20} = \frac{36}{25x^{20}}$$

Step3: Take the square root of the simplified expression

The square root of a fraction is the fraction of square roots. For non-negative terms, $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$, and $\sqrt{x^{2n}} = x^n$ for $x
eq0$:
$$\sqrt{\frac{36}{25x^{20}}} = \frac{\sqrt{36}}{\sqrt{25} \cdot \sqrt{x^{20}}} = \frac{6}{5x^{10}}$$

Answer:

$\frac{6}{5x^{10}}$ (corresponding to the first option)