Chapter A.1, Problem 32E

Solution

Simplify by removing factors from the radicand.

  $-3x{y}^2$ .

Given:

The given expression is $\sqrt[3]{-27x^3{y}^6}$ .

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 $\sqrt[3]{-27x^3{y}^6}$ .

Let $a=\sqrt[3]{-27x^3{y}^6}$

  $\begin{array}{l}$ a={(-27x^3{y}^6)}^{1/3}$a={\left((-3)\cdot (-3)\cdot (-3)\cdot x^3\cdot y^3\cdot y^3\right)}^{1/3}\\ a={\left({(-3)}^3\cdot x^3\cdot y^3\cdot y^3\right)}^{1/3}\\ a=-3\cdot x\cdot y\cdot y\\ a=-3x{y}^2\end{array}$

Thus the solution of given expression $\sqrt[3]{-27x^3{y}^6}$ is $-3x{y}^2$ .