Question

Write $6^{\frac{1}{4}},2^{\frac{1}{2}},4^{\frac{1}{3}}$ in ascending order.

Answer 1

Make the powers equal

We have $6^{\frac{1}{4}},2^{\frac{1}{2}},4^{\frac{1}{3}}$

L.C.M$\left(4,2,3\right)=12$ so multiply power in each exponent by $3,6,4$ respectively so that powers become equal.

So,$6^{\frac{3}{12}},2^{\frac{6}{12}},4^{\frac{4}{12}}$

It can be written as $6^{3\times \frac{1}{12}},2^{6\times \frac{1}{12}},4^{4\times \frac{1}{12}}$

$\Rightarrow {\left(216\right)}^{\frac{1}{12}},{\left(64\right)}^{\frac{1}{12}},{\left(256\right)}^{\frac{1}{12}}$

$\Rightarrow {\left(64\right)}^{\frac{1}{12}}<{\left(216\right)}^{\frac{1}{12}}<{\left(256\right)}^{\frac{1}{12}}$

Therefore the ascending order we have $2^{\frac{1}{2}}<6^{\frac{1}{4}}<4^{\frac{1}{3}}$