Chapter A.1, Problem 42E

Solution

Rationalize the denominator.

  $\frac{a^{3/5}\cdot b^{3/5}}{b}$ .

Given:

The given expression is $\sqrt[5]{\frac{a^3}{b^2}}$ .

Concept Used:

• Multiply numerator and denominator by a radical that will get rid of the radical in the denominator.
• Make sure all radicals are simplified.
• Simplify the fraction if needed.

Calculation:

The given expression is $\sqrt[5]{\frac{a^3}{b^2}}$ .

Multiply the given expression $\sqrt[5]{\frac{a^3}{b^2}}$ both numerator and denominator by $\sqrt[5]{b^3}$ .

  $\begin{array}{c}\sqrt[5]{\frac{a^3}{b^2}}=\frac{\sqrt[5]{a^3}}{\sqrt[5]{b^2}}\cdot \frac{\sqrt[5]{b^3}}{\sqrt[5]{b^3}}\\ =\frac{\sqrt[5]{a^3}\cdot \sqrt[5]{b^3}}{\sqrt[5]{b^5}}\\ =\frac{{(a^3\cdot b^3)}^{1/5}}{b}\\ =\frac{a^{3/5}\cdot b^{3/5}}{b}\end{array}$

Thus after rationalize the given expression $\sqrt[5]{\frac{a^3}{b^2}}$ is $\frac{a^{3/5}\cdot b^{3/5}}{b}$ .