Question

Find the factor which will rationalise:
$2+\sqrt[4]{47}$

Answer 1

We need to find the factor which when multiplied by $(2+\sqrt[4]{47})$ will lead to a rational number

Let $x=2$ and $y=7^{\frac{1}{4}}$, then $x^4$ and $y^4)$ are both rational

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

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

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

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

$\Rightarrow 2^3-2^2\cdot 7^{\frac{1}{4}}+2\cdot 7^{\frac{1}{2}}-7^{\frac{3}{4}}$