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Is the equation (4√3)24(3√)6=(43)5

true or false? Select the response that correctly answers the question and provides the appropriate justification.(1 point)
Responses

False. The Quotient Property has not been applied correctly.
False. The Quotient Property has not been applied correctly.

True. The Quotient Property has been applied correctly.
True. The Quotient Property has been applied correctly.

False. The Power of a Power Property has not been applied correctly.
False. The Power of a Power Property has not been applied correctly.

True. The Power of a Power Property has been applied correctly.

1 answer

To determine whether the equation \((4\sqrt{3})^{24}(3\sqrt{6}) = (4^3)^5\) is true, we can simplify both sides of the equation.

  1. Left Side:
    • \((4\sqrt{3})^{24} = (4 \cdot 3^{1/2})^{24} = 4^{24} \cdot (3^{1/2})^{24} = 4^{24} \cdot 3^{12}\)
    • \((3\sqrt{6})^{6} = (3 \cdot 6^{1/2})^{6} = 3^6 \cdot (6^{1/2})^{6} = 3^6 \cdot 6^{3} = 3^6 \cdot (2 \cdot 3)^{3} = 3^6 \cdot 2^{3} \cdot 3^{3} = 2^3 \cdot 3^{9}\)

Combining the left side gives us:

  • \(4^{24} \cdot 3^{12} \cdot 2^3 \cdot 3^{9} = 4^{24} \cdot 2^3 \cdot 3^{21}\)
  1. Right Side:
    • \((4^3)^5 = 4^{15}\)

Now we can compare both sides:

  • Left Side: \(4^{24} \cdot 2^3 \cdot 3^{21}\)
  • Right Side: \(4^{15}\)

Final Analysis: The left side clearly does not equal the right side based on the exponents of 4 alone, and since the left side has additional factors, we conclude that the equation is False.

Thus, the correct response is:

False. The Quotient Property has not been applied correctly.