Alright, let's go through the expression step by step.
Given the expression:
[tex]\[
\left[\left(x^n \right)^{n - \frac{1}{n}}\right]^{\frac{1}{n+1}}
\][/tex]
First, let's simplify the inner term:
[tex]\[
\left(x^n\right)^{n - \frac{1}{n}}
\][/tex]
To simplify this, we use the property of exponents [tex]\((a^{m})^{k} = a^{m \cdot k}\)[/tex]:
[tex]\[
\left(x^n \right)^{n - \frac{1}{n}} = x^{n \cdot \left(n - \frac{1}{n}\right)}
\][/tex]
Next, let's compute the exponent inside the parentheses:
[tex]\[
n \cdot \left(n - \frac{1}{n}\right) = n^2 - 1
\][/tex]
So we get:
[tex]\[
x^{n^2 - 1}
\][/tex]
Now, let's substitute this back into the original outer expression:
[tex]\[
\left(x^{n^2 - 1}\right)^{\frac{1}{n + 1}}
\][/tex]
Again, we apply the property of exponents [tex]\((a^{m})^{k} = a^{m \cdot k}\)[/tex]:
[tex]\[
x^{(n^2 - 1) \cdot \frac{1}{n + 1}}
\][/tex]
Thus, the final simplified expression becomes:
[tex]\[
x^{\frac{n^2 - 1}{n + 1}}
\][/tex]
So, the complete step-by-step simplification of the given expression is:
[tex]\[
\left[\left(x^n\right)^{n - \frac{1}{n}}\right]^{\frac{1}{n+1}} = x^{\frac{n^2 - 1}{n+1}}
\][/tex]
