Chapter 7.6, Problem 63PPS

To determine

Simplify the expression $\frac{8^{\frac{1}{6}}-9^{\frac{1}{4}}}{\sqrt{3}+\sqrt{2}}$ .

Answer to Problem 63PPS

The simplified value of expression $\frac{8^{\frac{1}{6}}-9^{\frac{1}{4}}}{\sqrt{3}+\sqrt{2}}$ is $-5+2^{\frac{3}{2}}{3}^{\frac{1}{2}}$ .

Explanation of Solution

Given information: The given expression is $\frac{8^{\frac{1}{6}}-9^{\frac{1}{4}}}{\sqrt{3}+\sqrt{2}}$ .

Calculation: The given expression $\frac{8^{\frac{1}{6}}-9^{\frac{1}{4}}}{\sqrt{3}+\sqrt{2}}$ can be simplified as,

  $\begin{array}{c}\frac{8^{\frac{1}{6}}-9^{\frac{1}{4}}}{\sqrt{3}+\sqrt{2}}=\frac{8^{\frac{1}{6}}-9^{\frac{1}{4}}}{\sqrt{3}+\sqrt{2}}\times \frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}-\sqrt{2}}\mathrm{......}\left(\text{Rationalized}\right)\\ =\frac{\left(8^{\frac{1}{6}}-9^{\frac{1}{4}}\right)\left(\sqrt{3}-\sqrt{2}\right)}{{\left(\sqrt{3}\right)}^2-{\left(\sqrt{2}\right)}^2}\\ =\frac{\left({\left(2^3\right)}^{\frac{1}{6}}-{\left(3^2\right)}^{\frac{1}{4}}\right)\left({\left(3\right)}^{\frac{1}{2}}-{\left(2\right)}^{\frac{1}{2}}\right)}{3-2}\\ =\frac{\left(2^{\frac{1}{2}}-3^{\frac{1}{2}}\right)\left(3^{\frac{1}{2}}-2^{\frac{1}{2}}\right)}{1}\end{array}$

Further simplified,

  $\begin{array}{c}\frac{8^{\frac{1}{6}}-9^{\frac{1}{4}}}{\sqrt{3}+\sqrt{2}}=-\frac{\left(3^{\frac{1}{2}}-2^{\frac{1}{2}}\right)\left(3^{\frac{1}{2}}-2^{\frac{1}{2}}\right)}{1}\\ =-{\left(3^{\frac{1}{2}}-2^{\frac{1}{2}}\right)}^2\\ =-\left[{\left(3^{\frac{1}{2}}\right)}^2-2\left(3^{\frac{1}{2}}\right)\left(2^{\frac{1}{2}}\right)+{\left(2^{\frac{1}{2}}\right)}^2\right]\\ =-\left[3-2^{\frac{3}{2}}{3}^{\frac{1}{2}}+2\right]\\ =-5+2^{\frac{3}{2}}{3}^{\frac{1}{2}}\end{array}$