To evaluate the equation \((\sqrt[8]{4})^{-24} \cdot (\sqrt{8})^{16} = 8^{-48}\), let's simplify both sides step by step.
First, we can rewrite the roots and exponents using properties of exponents:
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\(\sqrt[8]{4} = 4^{1/8}\), so \((\sqrt[8]{4})^{-24} = (4^{1/8})^{-24} = 4^{-24/8} = 4^{-3}\).
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Next, \(\sqrt{8} = 8^{1/2}\), so \((\sqrt{8})^{16} = (8^{1/2})^{16} = 8^{16/2} = 8^8\).
Now, substituting back into the equation:
\[ (\sqrt[8]{4})^{-24} \cdot (\sqrt{8})^{16} = 4^{-3} \cdot 8^8 \]
On the right side of the equation, we have \(8^{-48}\). Now simplify \(8^8\) as \(8^8 = (2^3)^8 = 2^{24}\).
Since \(4 = 2^2\), we can also rewrite \(4^{-3}\):
\[ 4^{-3} = (2^2)^{-3} = 2^{-6} \]
Now we have:
\[ 2^{-6} \cdot 8^8 = 2^{-6} \cdot 2^{24} = 2^{-6 + 24} = 2^{18} \]
Now consider \(8^{-48}\):
\[ 8^{-48} = (2^3)^{-48} = 2^{-144} \]
Now comparing both sides gives:
\[ 2^{18} \neq 2^{-144} \]
Therefore, we conclude that the equation is false.
So the correct answer is:
False, because the Negative Exponent Rule should be applied.
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