Chapter 6.6, Problem 39PPS

To determine

Simplify the expression $\frac{x^{\frac{1}{4}}+2}{x^{\frac{1}{4}}-2}$ .

Answer to Problem 39PPS

The simplified value of expression $\frac{x^{\frac{1}{4}}+2}{x^{\frac{1}{4}}-2}$ is $\frac{x^{\frac{1}{2}}+4x^{\frac{1}{4}}+4}{x^{\frac{1}{2}}-4}$ .

Explanation of Solution

Given information: The given expression is $\frac{x^{\frac{1}{4}}+2}{x^{\frac{1}{4}}-2}$ .

Calculation: The given expression $\frac{x^{\frac{1}{4}}+2}{x^{\frac{1}{4}}-2}$ can be simplified as,

  $\begin{array}{c}\frac{x^{\frac{1}{4}}+2}{x^{\frac{1}{4}}-2}=\frac{x^{\frac{1}{4}}+2}{x^{\frac{1}{4}}-2}\times \frac{x^{\frac{1}{4}}+2}{x^{\frac{1}{4}}+2}\mathrm{......}\left(\text{Rationalized}\right)\\ =\frac{{\left(x^{\frac{1}{4}}+2\right)}^2}{{\left(x^{\frac{1}{4}}\right)}^2-{\left(2\right)}^2}\mathrm{......}\left(\text{Simplified}\right)\\ =\frac{x^{\frac{1}{2}}+4x^{\frac{1}{4}}+4}{x^{\frac{1}{2}}-4}\end{array}$