Chapter A.10, Problem 41AYU

To determine

To find: Simplify the expression $\sqrt[3]{16x^{4}y}-3x\sqrt[3]{2xy}+5\sqrt[3]{-2x{y}^4}$.

Answer to Problem 41AYU

The value of $\sqrt[3]{16x^{4}y}-3x\sqrt[3]{2xy}+5\sqrt[3]{-2x{y}^4}=-\left(x+5y\right)\sqrt[3]{2xy}$.

Explanation of Solution

Given:

It is asked to find the simplest form of the given expression $\sqrt[3]{16x^{4}y}-3x\sqrt[3]{2xy}+5\sqrt[3]{-2x{y}^4}$.

Calculation:

Consider $\sqrt[3]{16x^{4}y}-3x\sqrt[3]{2xy}+5\sqrt[3]{-2x{y}^4}$,

$\sqrt[3]{16x^{4}y}-3x\sqrt[3]{2xy}+5\sqrt[3]{-2x{y}^4}=\sqrt[3]{8\times 2\times x^3\times x\times y}-3x\sqrt[3]{2xy}+5\sqrt[3]{\left(-1\right)2x{y}^3\times y}$

$=\sqrt[3]{2^3}\times \sqrt[3]{x^3}\times \sqrt[3]{2xy}-3x\sqrt[3]{2xy}+5\sqrt[3]{{\left(-1\right)}^3}\times \sqrt[3]{y^3}\times \sqrt[3]{2xy}$

$=2x\sqrt[3]{2xy}-3x\sqrt[3]{2xy}-5y\sqrt[3]{2xy}$

$=\sqrt[3]{2xy}\left(2x-3x-5y\right)$

$=-\left(x+5y\right)\sqrt[3]{2xy}$