To solve the given expression [tex]\(\left[\left(x^{a-a^{-1}}\right)^{\frac{1}{a-1}}\right]^{\frac{a}{a+1}}\)[/tex], we will simplify it step-by-step.
1. Start with the innermost expression:
[tex]\[
\left(x^{a - a^{-1}}\right)
\][/tex]
2. This expression is raised to the power of [tex]\(\frac{1}{a - 1}\)[/tex]:
[tex]\[
\left(x^{a - a^{-1}}\right)^{\frac{1}{a - 1}}
\][/tex]
When raising a power to another power, we multiply the exponents. So, the expression can be written as:
[tex]\[
x^{(a - a^{-1}) \cdot \frac{1}{a - 1}}
\][/tex]
3. We now have this in simpler form:
[tex]\[
x^{\frac{a - a^{-1}}{a - 1}}
\][/tex]
4. This entire expression is then raised to the power of [tex]\(\frac{a}{a + 1}\)[/tex]:
[tex]\[
\left(x^{\frac{a - a^{-1}}{a - 1}}\right)^{\frac{a}{a + 1}}
\][/tex]
5. Again, raising a power to another power, we multiply the exponents:
[tex]\[
x^{\left(\frac{a - a^{-1}}{a - 1}\right) \cdot \frac{a}{a + 1}}
\][/tex]
Hence, we can combine the exponents:
[tex]\[
x^{\left(\frac{a - a^{-1}}{a - 1} \cdot \frac{a}{a + 1}\right)}
\][/tex]
Combining and simplifying the exponents, the full exponent expression is:
[tex]\[
x^{\left(\frac{(a - a^{-1}) \cdot a}{(a - 1) \cdot (a + 1)}\right)}
\][/tex]
Upon simplifying, the term inside the final exponent simplifies further and specifically evaluates, yielding:
[tex]\[
\left[\left(x^{a - a^{-1}}\right)^{\frac{1}{a - 1}}\right]^{\frac{a}{a + 1}} = x
\][/tex]
Thus, the simplified form of the given expression is:
[tex]\(\boxed{x}\)[/tex]
