To determine whether the equation \( (4\sqrt{3})^{24} (3\sqrt{})^6 = (43)^5 \) is true or false, we need to analyze each side of the equation and examine the properties of exponents.
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Left Side: \( (4\sqrt{3})^{24} (3\sqrt{})^6 \):
- The term \( 4\sqrt{3} \) can be rewritten using fractional exponents as \( 4 \cdot 3^{1/2} \).
- This means \( (4\sqrt{3})^{24} = (4)^{24} \cdot (3^{1/2})^{24} = 4^{24} \cdot 3^{12} \).
- The term \( (3\sqrt{})^6 \) seems to have a typographical error. Assuming it meant \( (3\sqrt{3})^6 \), it can be rewritten as \( (3)^{6} \cdot (3^{1/2})^{6} = 3^{6} \cdot 3^{3} = 3^{9} \).
- Finally, combining both terms: \( 4^{24} \cdot 3^{12} \cdot 3^{9} = 4^{24} \cdot 3^{21} \).
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Right Side: \( (43)^5 \):
- This means \( (4 \cdot 3)^5 = 4^{5} \cdot 3^{5} \).
Now, the two sides are:
- Left Side: \( 4^{24} \cdot 3^{21} \)
- Right Side: \( 4^{5} \cdot 3^{5} \)
At this point, we can see that the exponents of the bases \( 4 \) and \( 3 \) are vastly different between the two sides.
Conclusion: The equation is False. The appropriate reasoning is that the properties of exponents have not been applied correctly in relating the terms. Therefore, the correct response is:
False. The Quotient Property has not been applied correctly. (This choice seems to imply mistakes with exponent application, but the essence is ongoing inaccuracies in both terms explored.)
5 answers