Is the equation (root(8, 4)) ^ - 24 * (sqrt(8)) ^ 16 = 8 ^ - 48 answers the question justification.

To evaluate whether the equation \((\sqrt[8]{4})^{-24} \cdot (\sqrt{8})^{16} = 8^{-48}\) is true or false, we can simplify each side of the equation step by step.

1. **Simplifying the left side:**

- First, let's rewrite \(\sqrt[8]{4}\) and \(\sqrt{8}\) in terms of exponentials:
\[
\sqrt[8]{4} = 4^{1/8}
\]
\[
\sqrt{8} = 8^{1/2}
\]

- Now substitute these into the left side:
\[
(4^{1/8})^{-24} \cdot (8^{1/2})^{16}
\]
- Applying the power of a power rule \((a^m)^n = a^{m \cdot n}\):
\[
4^{-24/8} \cdot 8^{16/2} = 4^{-3} \cdot 8^8
\]

- Next, we can express \(4^{x}\) and \(8^{y}\) with base 2:
\[
4 = 2^2 \quad \Rightarrow \quad 4^{-3} = (2^2)^{-3} = 2^{-6}
\]
\[
8 = 2^3 \quad \Rightarrow \quad 8^8 = (2^3)^8 = 2^{24}
\]

- Therefore, the left side now becomes:
\[
2^{-6} \cdot 2^{24} = 2^{-6 + 24} = 2^{18}
\]

2. **Simplifying the right side:**
\[
8^{-48} = (2^3)^{-48} = 2^{-144}
\]

Now we have:

\[
2^{18} \neq 2^{-144}
\]

So, the two sides are not equal, thus the original equation is false.

Now, let's analyze the provided responses to choose the correct one:

1. **When the bases are multiplied, the exponents are also multiplied.**
- This statement is misleading since we didn't multiply bases here, so this is not applicable.

2. **True, but the equation can be simplified further.**
- False, because the two sides don't equal after simplification.

3. **False, because the Negative Exponent Rule should be applied.**
- This isn't the main reason the equation is false. While the negative exponent rule applies to the left side, it is not the sole issue with the original equation.

4. **False, because the exponents should be added rather than multiplied.**
- This is also misleading as we didn't need to add exponents in this situation.

Based on the analysis, the equation is indeed false, but there isn't a perfectly accurate justification among the responses. However, if you had to choose one, **"False, because the Negative Exponent Rule should be applied."** can be selected because it indicates something is incorrect, but it's not entirely the full reasoning.

So the final answer is **False** and due to lack of the most accurate explanation among the choices, the best choice reflecting any error would be the third one.