Question

Express with rational denominator:
$\frac{\sqrt[6]{69}-\sqrt[6]{68}}{\sqrt[6]{69}+\sqrt[6]{68}}$

Answer 1

We have $\frac{\sqrt[6]{69}-\sqrt[6]{68}}{\sqrt[6]{69}+\sqrt[6]{68}}=\frac{\sqrt[3]{33}-\sqrt[2]{22}}{\sqrt[3]{33}+\sqrt[2]{22}}$

We need to rationalise the denominator which is $\sqrt[3]{33}+\sqrt[2]{22}$.

Let $x=3^{\frac{1}{3}}$ and $y=2^{\frac{1}{2}}$

Since, $x^6-y^6=(x+y)(x^5-x^{4}y+x^3{y}^2-x^2{y}^3+x{y}^4-y^5)$

Hence, the required factor is $3^{\frac{5}{3}}-3^{\frac{4}{3}}\cdot 2^{\frac{1}{2}}+3^{\frac{3}{3}}\cdot 2^{\frac{2}{2}}-3^{\frac{2}{3}}\cdot 2^{\frac{3}{2}}+3^{\frac{1}{3}}\cdot 2^{\frac{4}{2}}-2^{\frac{5}{2}}$

Therefore, the given expression becomes $\frac{(3^{\frac{1}{3}}-2^{\frac{1}{2}})(3^{\frac{5}{3}}-3^{\frac{4}{3}}\cdot 2^{\frac{1}{2}}+3^{\frac{3}{3}}\cdot 2^{\frac{2}{2}}-3^{\frac{2}{3}}\cdot 2^{\frac{3}{2}}+3^{\frac{1}{3}}\cdot 2^{\frac{4}{2}}-2^{\frac{5}{2}})}{1}$.

$=17-3^{\frac{5}{3}}\cdot 2^{\frac{3}{2}}+3^{\frac{4}{3}}\cdot 2^2-3\cdot 2^{\frac{5}{2}}+3^{\frac{2}{3}}\cdot 2^3-3^{\frac{1}{3}}\cdot 2^{\frac{7}{2}}$