Value of `[(x^l)^(1 - 1/l)]^(1/(l - 1))` is:

Options

A. a. `x`
B. b. `1`
C. c. `x^(-1)`
D. d. `x^(-l)`

Answer

To evaluate the expression \(\left[(x^l)^{(1 - \frac{1}{l})}\right]^{\frac{1}{l - 1}}\), we can follow these steps:

### Step 1: Simplify the inner exponent
We start with the expression:
\[ \left[(x^l)^{(1 - \frac{1}{l})}\right]^{\frac{1}{l - 1}} \]

First, simplify the exponent \(1 - \frac{1}{l}\):
\[ 1 - \frac{1}{l} = \frac{l - 1}{l} \]

### Step 2: Rewrite the expression
Now substitute this back into the expression:
\[ \left[(x^l)^{\frac{l - 1}{l}}\right]^{\frac{1}{l - 1}} \]

### Step 3: Apply the power of a power property
Using the property of exponents \((a^m)^n = a^{m \cdot n}\), we can rewrite the expression:
\[ (x^l)^{\frac{l - 1}{l} \cdot \frac{1}{l - 1}} = (x^l)^{\frac{1}{l}} \]

### Step 4: Simplify further
Now simplify \((x^l)^{\frac{1}{l}}\):
\[ x^{l \cdot \frac{1}{l}} = x^1 = x \]

### Final Result
Thus, the value of the expression \(\left[(x^l)^{(1 - \frac{1}{l})}\right]^{\frac{1}{l - 1}}\) is:
\[ \boxed{x} \]