To determine whether the equation \( (4\sqrt{3})^{24}(3\sqrt{6})^{6} = (4^{3})^{5} \) is true or false, we'll need to analyze both sides.
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Left Side:
- \( (4\sqrt{3})^{24} = (4 \cdot 3^{1/2})^{24} = 4^{24} \cdot (3^{1/2})^{24} = 4^{24} \cdot 3^{12} \)
- \( (3\sqrt{6})^{6} = (3 \cdot 6^{1/2})^{6} = 3^{6} \cdot (6^{1/2})^{6} = 3^{6} \cdot 6^{3} = 3^{6} \cdot (6 = 2 \cdot 3)^{3} = 3^{6} \cdot 2^{3} \cdot 3^{3} = 2^{3} \cdot 3^{9} \)
Now adding these results together: \[ (4\sqrt{3})^{24}(3\sqrt{6})^{6} = (4^{24} \cdot 3^{12}) \cdot (2^{3} \cdot 3^{9}) = 4^{24} \cdot 2^{3} \cdot 3^{12 + 9} = 4^{24} \cdot 2^{3} \cdot 3^{21} \]
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Right Side:
- \( (4^{3})^{5} = 4^{3 \cdot 5} = 4^{15} \)
Now let's compare the two sides:
- Left Side: \( 4^{24} \cdot 2^{3} \cdot 3^{21} \)
- Right Side: \( 4^{15} \)
Clearly, \( 4^{24} \) and \( 4^{15} \) are not equal, and so the left side does not equal the right side.
Thus, the correct response is:
False. The Power of a Power Property has not been applied correctly.
Since the left side simplifies to a different expression than the right side, the equation is false.