To determine whether the equation \((4\sqrt{3})^{24}(3\sqrt{})^{6}=(4^{3})^{5}\) is true or false, we will analyze both sides step by step.
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Left Side: \( (4\sqrt{3})^{24}(3\sqrt{})^{6} \)
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First, rewrite \( \sqrt{3} \) as \( 3^{1/2} \): \[ (4 \cdot 3^{1/2})^{24} \]
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Apply the Power of a Power Property: \[ 4^{24} (3^{1/2})^{24} = 4^{24} \cdot 3^{12} \]
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For \( (3\sqrt{})^{6} \), if we treat it as \( 3^{6} \) (assuming this was the intended form):
- This part is straightforward since it is \( 3^{6} \).
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Thus, the full left side simplifies to: \[ 4^{24} \cdot 3^{12} \cdot 3^{6} = 4^{24} \cdot 3^{18} \]
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Right Side: \( (4^{3})^{5} \)
- Again, apply the Power of a Power Property: \[ 4^{3 \cdot 5} = 4^{15} \]
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Comparison of Both Sides:
- Left Side: \( 4^{24} \cdot 3^{18} \)
- Right Side: \( 4^{15} \)
Clearly, \( 4^{24} \cdot 3^{18} \neq 4^{15} \).
Thus, the statement is False, and the correct justification is: False. The Power of a Power Property has not been applied correctly.