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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.
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