Question

Find the factor which will rationalise:
$\sqrt[6]{65}+\sqrt[3]{32}$

Answer 1

We need to find the factor which when multiplied by $(\sqrt[6]{65}+\sqrt[3]{32})$ will lead to a rational number

Let $x=5^{\frac{1}{6}}$ and $y=2^{\frac{1}{3}}$, then $x^6$ and $y^6$ are both rational

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

Hence, $x^5-x^{4}y+x^3{y}^2-x^2{y}^3+x{y}^4-y^5$ is the required factor.

Substituting the value for $x$ and $y$, then the required factor is

$5^{\frac{5}{6}}-5^{\frac{4}{6}}\cdot 2^{\frac{1}{3}}+5^{\frac{3}{6}}\cdot 2^{\frac{2}{3}}-5^{\frac{2}{6}}\cdot 2^{\frac{3}{3}}+5^{\frac{1}{6}}\cdot 2^{\frac{4}{3}}-2^{\frac{5}{3}}$

$\Rightarrow 5^{\frac{5}{6}}-5^{\frac{4}{6}}\cdot 2^{\frac{1}{3}}+5^{\frac{3}{6}}\cdot 2^{\frac{2}{3}}-5^{\frac{2}{6}}\cdot 2+5^{\frac{1}{6}}\cdot 2^{\frac{4}{3}}-2^{\frac{5}{3}}$