what is the simplified form of $sqrt{\frac{72x^{16}}{50x^{36}}}$? assume $x\neq0$.\n$\frac{6}{5x^{10}}$\n$\fr…

Explanation:

Step1: Simplify the coefficient fraction

Simplify $\frac{72}{50}$ to $\frac{36}{25}$ by dividing both numerator and denominator by 2.

Step2: Use exponent - division rule

For the $x$ terms, use $\frac{x^{m}}{x^{n}}=x^{m - n}$. So $\frac{x^{16}}{x^{36}}=x^{16-36}=x^{- 20}$.

Step3: Combine and apply square - root

The expression becomes $\sqrt{\frac{36}{25}x^{-20}}$. Since $\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}$ ($a = \frac{36}{25}$, $b=x^{-20}$), we have $\sqrt{\frac{36}{25}}\cdot\sqrt{x^{-20}}$. And $\sqrt{\frac{36}{25}}=\frac{6}{5}$, $\sqrt{x^{-20}}=x^{-10}=\frac{1}{x^{10}}$.

Answer:

$\frac{6}{5x^{10}}$