`{(-2)^((-2))}^((-2))` is equal to :

Options

A. a) `16`
B. b) `8`
C. c) `-8`
D. d) `-1`

Answer

To solve the expression \({(-2)}^{(-2)^{(-2)}}\), we will follow the laws of exponents step by step.

### Step-by-Step Solution:

1. **Identify the Expression**:
We start with the expression:
\[ {(-2)}^{(-2)^{(-2)}} \]

2. **Apply the Law of Exponents**:
According to the law of exponents, when we have a power raised to another power, we multiply the exponents:
\[ a^{m^n} = a^{m \cdot n} \]
Here, we can rewrite our expression as:
\[ {(-2)}^{(-2)^{(-2)}} = {(-2)}^{(-2) \cdot (-2)} \]

3. **Calculate the Exponent**:
Now, we calculate the exponent:
\[ (-2) \cdot (-2) = 4 \]
So, we can rewrite the expression as:
\[ {(-2)}^{4} \]

4. **Evaluate the Power**:
Now we need to evaluate \({(-2)}^{4}\):
\[ {(-2)}^{4} = (-2) \cdot (-2) \cdot (-2) \cdot (-2) \]
We can calculate this step by step:
- First, \((-2) \cdot (-2) = 4\)
- Next, \(4 \cdot (-2) = -8\)
- Finally, \(-8 \cdot (-2) = 16\)

Thus, \({(-2)}^{4} = 16\).

5. **Final Result**:
Therefore, the value of the original expression \({(-2)}^{(-2)^{(-2)}}\) is:
\[ \boxed{16} \]