Question

What we should multiply with $\frac{\sqrt{x^3}\sqrt[3]{3x^5}}{\sqrt[5]{5x^3}}.\sqrt[30]{30x^{77}}$ to get $1$

Answer 1

$\frac{\sqrt{x^3}.\sqrt[3]{3x^5}}{\sqrt[5]{5x^3}}.\sqrt[30]{30x^{77}}\times \left(K\right)=1$

we have to find K.

$\Rightarrow \frac{x^{\frac{3}{2}}.x^{\frac{5}{3}}}{x^{\frac{3}{5}}}.x^{\frac{77}{30}}.K=1$

$\Rightarrow x^{[\frac{3}{2}+\frac{5}{3}+\frac{77}{30}-\frac{3}{5}]}\times K=1$

$\Rightarrow x^{\left[\frac{45+50+77-18}{30}\right]}\times K=1$

$\Rightarrow x^{\left[\frac{172-18}{30}\right]}\times K=1$

$\Rightarrow x^{\left(\frac{154}{30}\right)}\times K=1$

$\Rightarrow K=x^{\frac{-154}{30}}$

$\Rightarrow K=x^{\frac{-77}{15}}$

$\Rightarrow K=\left(\sqrt[15]{15x^{-77}}\right)$