To evaluate the equation \(\frac{(\sqrt[4]{3})^{24}}{(\sqrt{3})^6} = \left(\frac{4}{3}\right)^5\), let's first simplify the left-hand side.
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Evaluate \(\sqrt[4]{3}\): \(\sqrt[4]{3} = 3^{1/4}\)
So, \((\sqrt[4]{3})^{24} = (3^{1/4})^{24} = 3^{(1/4) \cdot 24} = 3^6\).
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Evaluate \(\sqrt{3}\): \(\sqrt{3} = 3^{1/2}\)
So, \((\sqrt{3})^6 = (3^{1/2})^6 = 3^{(1/2) \cdot 6} = 3^3\).
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Now substitute these values back into the left-hand side: \[ \frac{3^6}{3^3} = 3^{6-3} = 3^3 \]
The left-hand side simplifies to \(3^3\).
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Now evaluate the right-hand side: \[ \left(\frac{4}{3}\right)^5 = \frac{4^5}{3^5} \]
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Comparing both sides: We have:
- Left-hand side: \(3^3\)
- Right-hand side: \(\frac{4^5}{3^5}\)
To compare these, let's also express \(3^3\) using a common denominator: \[ 3^3 = \frac{27}{1} = \frac{27 \cdot 3^5}{3^5} = \frac{27 \cdot 3^5}{3^5} = \frac{27}{3^5} \]
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Now we compare \(27\) and \(4^5\): We know \(4^5 = 1024\), so \(\frac{27}{3^5}\) is not equal to \(\frac{1024}{3^5}\).
Thus, since \(3^3 \neq \left(\frac{4}{3}\right)^5\), the original equation is false.
Final Answer: False. The Power of a Power Property has not been applied correctly.
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