Question

The equation (^4 square root 8)^-24•(square root 8)^16=8^-48 is false the equation is false because the exponents should be_____ rather than ____

Answer 1

To analyze the equation \((\sqrt[4]{8})^{-24} \cdot (\sqrt{8})^{16} = 8^{-48}\), we start by rewriting the terms with exponents:

1. Rewrite \(\sqrt[4]{8}\):
\[
\sqrt[4]{8} = 8^{1/4}
\]
Therefore,
\[
(\sqrt[4]{8})^{-24} = (8^{1/4})^{-24} = 8^{-24/4} = 8^{-6}
\]

2. Rewrite \(\sqrt{8}\):
\[
\sqrt{8} = 8^{1/2}
\]
Therefore,
\[
(\sqrt{8})^{16} = (8^{1/2})^{16} = 8^{16/2} = 8^8
\]

Now, we can combine the results:
\[
(\sqrt[4]{8})^{-24} \cdot (\sqrt{8})^{16} = 8^{-6} \cdot 8^{8}
\]

Since we add the exponents when multiplying the same bases:
\[
8^{-6 + 8} = 8^{2}
\]

Now, we compare this to the right side of the original equation \(8^{-48}\):
\[
8^{2} \neq 8^{-48}
\]

Thus, the equation is indeed false because:

**The exponents should be \(2\) rather than \(-48\)**.