Chapter A.1, Problem 69E

Solution

Simplify the radical expression.

  $\frac{2x}{y}\sqrt[3]{x}$ .

Given:

The given expression is $\sqrt[3]{\frac{4x^2}{y^2}}\cdot \sqrt[3]{\frac{2x^2}{y}}$ .

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 $\sqrt[3]{\frac{4x^2}{y^2}}\cdot \sqrt[3]{\frac{2x^2}{y}}$ .

  $\begin{array}{c}\sqrt[3]{\frac{4x^2}{y^2}}\cdot \sqrt[3]{\frac{2x^2}{y}}=\sqrt[3]{\frac{8x^2{x}^2}{y^{2}y}}\\ =\sqrt[3]{\frac{2^3{x}^4}{y^3}}\\ =\sqrt[3]{\frac{2^3{x}^{3}x}{y^3}}\\ =\frac{2x}{y}\sqrt[3]{x}\end{array}$

Thus after simplifying $\sqrt[3]{\frac{4x^2}{y^2}}\cdot \sqrt[3]{\frac{2x^2}{y}}$ we get $\frac{2x}{y}\sqrt[3]{x}$ .