To determine whether the equation \((\sqrt[8]{4})^{-24} \cdot (\sqrt{8})^{16} = 8^{-48}\) is valid, let's simplify both sides.
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Simplifying the Left Side:
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\(\sqrt[8]{4}\) can be expressed as \(4^{1/8}\).
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Then, \((\sqrt[8]{4})^{-24} = (4^{1/8})^{-24} = 4^{-24/8} = 4^{-3}\).
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\(\sqrt{8}\) can be expressed as \(8^{1/2}\).
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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: \[ 2^{-6} \cdot 2^{24} = 2^{-6 + 24} = 2^{18} \]
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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.
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