Proof of Given Expression = x
Question
\[
\left\{(x^{a-1})^{\frac{1}{a-1}}\right\}^{\frac{a}{a+1}}
\]
Solution
\[
= (x^{a-1})^{\frac{1}{a-1} \cdot \frac{a}{a+1}}
\]
\[
= x^{\frac{a}{a+1}}
\]
\[
= x
\]
Answer
\[
\boxed{x}
\]
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