Chapter A.1, Problem 39E

Solution

Rationalize the denominator.

  $\frac{x^{3/5}}{x}$ .

Given:

The given expression is $\frac{1}{\sqrt[5]{x^2}}$ .

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 $\frac{1}{\sqrt[5]{x^2}}$ .

Multiply the given expression $\frac{1}{\sqrt[5]{x^2}}$ both numerator and denominator by $\sqrt[5]{x^3}$ .

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

Thus after rationalize the given expression $\frac{1}{\sqrt[5]{x^2}}$ is $\frac{x^{3/5}}{x}$ .