To analyze the equation \((4\sqrt{3})^{24}(3\sqrt{})^{6} = (4^{3})^{5}\), we must use properties of exponents.
Let's break down each side of the equation.
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Left Side: \((4\sqrt{3})^{24}(3\sqrt{})^{6}\)
- The term \(4\sqrt{3}\) can be rewritten as \(4 \cdot 3^{1/2}\).
- Thus, \((4\sqrt{3})^{24} = 4^{24} \cdot (3^{1/2})^{24} = 4^{24} \cdot 3^{12}\).
- The term \(3\sqrt{} \) seems like it might be a typo, and I'm assuming you meant \(3^{1/2}\) repeated with the exponent of 6.
- So, if we take \(3\sqrt{3}\), we can interpret it, but it lacks clarity.
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Right Side: \((4^{3})^{5}\)
- Using the Power of a Power Property, we can rewrite this as \(4^{3 \times 5} = 4^{15}\).
So now we need to compare the left and right side values systematically.
- The left-hand side expanded gives us a combination involving both \(4\) raised to a higher power and \(3\) raised to a power. However, we should focus where the powers land:
- As it stands, if \(3^{6}\) is included in further total comparisons and \(4^{24}\) remains strong.
From the observation, after careful review and clarification, the evaluation of both reduces down to confirming if both ends can match in powers.
Conclusion:
Given the confusion in the interpretation of the terms, initially, it seems the equation as posited could not restate equivalently as initially anticipated. If we assume the multiplication was not applied correctly or a misrepresentation existed initially inspector for error signs.
The answer is: False. The Power of a Power Property has not been applied correctly.
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