Show that:$ \left\{\left(x^{a-a^{-1}}\right)^{\frac{1}{a-1}}\right\}^{\frac{a}{a+1}}=x $

To do: 

We have to show that \( \left\{\left(x^{a-a^{-1}}\right)^{\frac{1}{a-1}}\right\}^{\frac{a}{a+1}}=x \).

Solution:

We know that,

$(a^{m})^{n}=a^{m n}$

$a^{m} \times a^{n}=a^{m+n}$

$a^{m} \div a^{n}=a^{m-n}$

$a^{0}=1$

Therefore,

LHS $=[{(x^{a-a^{-1}})^{\frac{1}{a-1}}}]^{\frac{a}{a+1}}$

$=[{(x^{a-\frac{1}{a}})^{\frac{1}{a-1}}}]^{\frac{a}{a+1}}$

$=x^{\frac{a^{2}-1}{a} \times \frac{1}{a-1} \times \frac{a}{a+1}}$

$=x^{\frac{(a+1)(a-1)}{a} \times \frac{1}{a-1} \times \frac{a}{a+1}}$

$=x^{1}$

$=x$

$=$ RHS

Hence proved.