What is the` (m-1)^(th)" root of "[(a^(m))^(m)-(1/m)]^(1/(m+1))` ?

Options

A. `a^(m+(1//m))`
B. `a^(m-(1//m))`
C. a
D. 1

Explanation

To solve the problem of finding the \((m-1)^{th}\) root of \(\left[(a^{m})^{m} - \frac{1}{m}\right]^{\frac{1}{m+1}}\), we will follow these steps:

### Step 1: Simplify the expression inside the root
We start with the expression:
\[ \left[(a^{m})^{m} - \frac{1}{m}\right]^{\frac{1}{m+1}} \]
This can be rewritten as:
\[ \left[a^{m^2} - \frac{1}{m}\right]^{\frac{1}{m+1}} \]

### Step 2: Apply the \((m-1)^{th}\) root
Now we need to take the \((m-1)^{th}\) root of the expression:
\[ \left[a^{m^2} - \frac{1}{m}\right]^{\frac{1}{m+1}}^{\frac{1}{m-1}} \]
This can be simplified using the property of exponents:
\[ \left[a^{m^2} - \frac{1}{m}\right]^{\frac{1}{(m+1)(m-1)}} \]

### Step 3: Combine the exponents
Now we can express the exponent as:
\[ \frac{1}{(m+1)(m-1)} \]
So we have:
\[ \left[a^{m^2} - \frac{1}{m}\right]^{\frac{1}{(m+1)(m-1)}} \]

### Step 4: Final expression
Thus, the final expression for the \((m-1)^{th}\) root of the original expression is:
\[ \left[a^{m^2} - \frac{1}{m}\right]^{\frac{1}{(m+1)(m-1)}} \]