Question

Is the equation (root(8, 4)) ^ - 24 * (sqrt(8)) ^ 16 = 8 ^ - 48 answers the question justification. (1 point) true or false? Select the response that correctly True. When the bases are multiplied, the exponents are also multiplied. True, but the equation can be simplified further. False, because the Negative Exponent Rule should be applied. False, because the exponents should be added rather than multiplied.

Answer 1

To determine whether the equation \((\sqrt[8]{4})^{-24} \cdot (\sqrt{8})^{16} = 8^{-48}\) is valid, let's simplify both sides.

1. **Simplifying the Left Side:**

- \(\sqrt[8]{4}\) can be expressed as \(4^{1/8}\).
- Then, \((\sqrt[8]{4})^{-24} = (4^{1/8})^{-24} = 4^{-24/8} = 4^{-3}\).

- \(\sqrt{8}\) can be expressed as \(8^{1/2}\).
- Thus, \((\sqrt{8})^{16} = (8^{1/2})^{16} = 8^{16/2} = 8^8\).

Now, combine these results:
\[
(\sqrt[8]{4})^{-24} \cdot (\sqrt{8})^{16} = 4^{-3} \cdot 8^8
\]

Next, express \(4^{-3}\) in terms of base \(8\):
\[
4 = 2^2 \Rightarrow 4^{-3} = (2^2)^{-3} = 2^{-6}
\]

Since \(8 = 2^3\), we can express \(8^8\) as:
\[
8^8 = (2^3)^8 = 2^{24}
\]

Therefore, the left side 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 compare both sides:
- Left Side: \(2^{18}\)
- Right Side: \(2^{-144}\)

Clearly, \(2^{18} \neq 2^{-144}\).

So the original equation is **false**.

In regards to the justification:
- The correct response is: **False, because the exponents should be added rather than multiplied.**

Thus, the final answer is **False** as the equation does not hold true.