The value of `( 16)^(1//4)` are

Options

A. `+- 2, +- 2i`
B. `+-4, +- 4i`
C. `+-1, +-i`
D. None of these

Answer

To solve the problem of finding the value of \( 16^{1/4} \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Expression**:
We start with the expression \( x = 16^{1/4} \).

2. **Rewrite the Expression**:
We can express 16 as a power of 2:
\[ 16 = 2^4 \]
Therefore, we can rewrite our expression:
\[ x = (2^4)^{1/4} \]

3. **Apply the Power of a Power Rule**:
Using the property of exponents \( (a^m)^n = a^{m \cdot n} \), we get:
\[ x = 2^{4 \cdot (1/4)} = 2^1 = 2 \]

4. **Consider the Roots**:
Since we are looking for all values of \( x \), we need to consider the equation \( x^4 = 16 \).
This can be rewritten as:
\[ x^4 - 16 = 0 \]

5. **Factor the Equation**:
We can factor this as a difference of squares:
\[ (x^2 - 4)(x^2 + 4) = 0 \]

6. **Solve Each Factor**:
- For \( x^2 - 4 = 0 \):
\[ x^2 = 4 \implies x = \pm 2 \]
- For \( x^2 + 4 = 0 \):
\[ x^2 = -4 \implies x = \pm \sqrt{-4} = \pm 2i \]

7. **Compile All Solutions**:
Thus, the complete set of solutions for \( x \) is:
\[ x = 2, -2, 2i, -2i \]

### Final Answer:
The values of \( 16^{1/4} \) are \( 2, -2, 2i, -2i \).

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