Chapter A.1, Problem 34E

Solution

Simplify by removing factors from the radicand.

  $2x^{2}y\sqrt[3]{y}$ .

Given:

The given expression is $\sqrt[3]{8x^6{y}^4}$ .

Concept Used:

• Find the largest factor in the radicand that is a perfect power of the index. Rewrite the radicand as a product of two factors, using that factor.
• Use the product rule to rewrite the radical as the product of two radicals.
• Simplify the root of the perfect power.

Calculation:

The given expression is $\sqrt[3]{8x^6{y}^4}$ .

Let $a=\sqrt[3]{8x^6{y}^4}$

  $\begin{array}{l}$ a={(8x^6{y}^4)}^{1/3}$a={\left(2\cdot 2\cdot 2\cdot x^3\cdot x^3\cdot y^3\cdot y\right)}^{1/3}\\ a=x\cdot x\cdot y\cdot 2{\left(y\right)}^{1/3}\\ a=2x^{2}y{\left(y\right)}^{1/3}\\ a=2x^{2}y\sqrt[3]{y}\end{array}$

Thus the solution of given expression $\sqrt[3]{8x^6{y}^4}$ is $2x^{2}y\sqrt[3]{y}$ .