Is the equation (\root(4)(8))^(-24)* (\sqrt(8))^(16) =8^(-48) true or false? select the response that correctly answers the question and provides the appropriate justification

To determine whether the equation \((\sqrt[4]{8})^{-24} \cdot (\sqrt{8})^{16} = 8^{-48}\) is true, we should simplify both sides of the equation.

Let's break down each term:

1. **Convert the expressions to powers of 8**:
- The fourth root can be expressed as an exponent: \(\sqrt[4]{8} = 8^{1/4}\). Thus, \((\sqrt[4]{8})^{-24} = (8^{1/4})^{-24} = 8^{-24/4} = 8^{-6}\).
- The square root can also be expressed similarly: \(\sqrt{8} = 8^{1/2}\). Therefore, \((\sqrt{8})^{16} = (8^{1/2})^{16} = 8^{16/2} = 8^8\).

2. **Combine both terms**:
- Now we have:
\[
8^{-6} \cdot 8^{8} = 8^{(-6 + 8)} = 8^2
\]

3. **Compare with the right side**:
- The right side of the equation is \(8^{-48}\).

Now we can see:
\[
8^2 \neq 8^{-48}
\]

Thus, the equation is **false**.

### Correct Response:
**false, bc the negative exponent rule should be applied**.

This justification is based on the simplification process and comparison of the resulting exponents, demonstrating that both sides of the equation are not equal.