`[1-[1-(1-a^(4))^(-1)]^(-1)]^(-1/4)` is equal to:

Options

A. `a^(4)`
B. `a^(2)`
C. a
D. `1/a`

Answer

To simplify the expression \([1-[1-(1-a^{4})^{-1}]^{-1}]^{-1/4}\), we will follow these steps:

### Step-by-Step Solution:

1. **Start with the innermost expression**:
\[ 1 - a^4 \]
This is the expression we will manipulate first.

2. **Apply the negative exponent**:
The expression becomes:
\[ (1 - a^4)^{-1} = \frac{1}{1 - a^4} \]

3. **Substitute back into the expression**:
Now we replace the innermost part in the original expression:
\[ 1 - \frac{1}{1 - a^4} \]

4. **Combine the fractions**:
To combine the fractions, we need a common denominator:
\[ 1 - \frac{1}{1 - a^4} = \frac{(1 - a^4) - 1}{1 - a^4} = \frac{-a^4}{1 - a^4} \]

5. **Substitute back into the expression**:
Now we have:
\[ 1 - \left(\frac{-a^4}{1 - a^4}\right)^{-1} \]

6. **Apply the negative exponent again**:
This becomes:
\[ 1 - \left(\frac{1 - a^4}{-a^4}\right) = 1 + \frac{1 - a^4}{a^4} \]

7. **Combine the fractions**:
Combine the fractions:
\[ 1 + \frac{1 - a^4}{a^4} = \frac{a^4 + 1 - a^4}{a^4} = \frac{1}{a^4} \]

8. **Final exponent application**:
Now we take the entire expression to the power of \(-\frac{1}{4}\):
\[ \left(\frac{1}{a^4}\right)^{-1/4} = a^{4 \cdot \frac{1}{4}} = a \]

Thus, the simplified expression is:
\[ \boxed{a} \]