Chapter A.10, Problem 17AYU

To determine

To find: Simplify the expression $\sqrt[4]{\frac{x^9{y}^7}{x{y}^3}}$.

Answer to Problem 17AYU

The value of $\sqrt[4]{\frac{x^9{y}^7}{x{y}^3}}=x^{2}y$.

Explanation of Solution

Given:

It is asked to find the simplest form of the given expression $\sqrt[4]{\frac{x^9{y}^7}{x{y}^3}}$.

Calculation:

If $n\ge 2$ is an integer and $a$ is a real number, then

$\sqrt[n]{a^n}=a$ if $n\ge 3$ is odd.

$\sqrt[n]{a^n}=\left|a\right|$ if $n\ge 2$ is even 9.

$\sqrt[n]{ab}=\sqrt[n]{a}\sqrt[n]{b}$

$\sqrt[4]{\frac{x^9{y}^7}{x{y}^3}}=\sqrt[4]{x^8\times y^4}=\sqrt[4]{{\left(x^2\right)}^4\times y^4}\\$

$\sqrt[4]{{\left(x^2\right)}^4\times y^4}=\sqrt[4]{{\left(x^2\right)}^4}\times \sqrt[4]{{\left(y\right)}^4}=\left|x^2\right|\times \left|y\right|$

From the given data, it can be seen that all the given variables are positive. Therefore,

$\left|x^2\right|\times \left|y\right|=x^{2}y$