Chapter 4.2, Problem 10QR

To determine

To Express: The given radicals into rational powers.

Answer to Problem 10QR

The required expression is $\frac{1}{x^{\frac{1}{2}}}+\frac{1}{x^{\frac{5}{6}}}$ .

Explanation of Solution

Given information: $\frac{x+\sqrt[3]{x^2}}{\sqrt{x^3}}$ .

Formula used: $\sqrt[n]{x}=x^{\frac{1}{n}},\frac{x^a}{x^b}=x^{a-b}$ .

Calculation:

  $\begin{array}{c}\frac{x+\sqrt[3]{x^2}}{\sqrt{x^3}}=\frac{x+x^{\frac{2}{3}}}{x^{\frac{3}{2}}}\\ =\frac{x}{x^{\frac{3}{2}}}+\frac{x^{\frac{2}{3}}}{x^{\frac{3}{2}}}\\ =x^{1-\frac{3}{2}}+x^{\frac{2}{3}-\frac{3}{2}}\\ =x^{\frac{-1}{2}}+x^{\frac{-5}{6}}\\ =\frac{1}{x^{\frac{1}{2}}}+\frac{1}{x^{\frac{5}{6}}}\end{array}$