To determine whether the equation \(3^{-5} \times (4\sqrt{81})^3 / 3^{-4} = 3^{-6}\) is true or false, let's simplify it step by step.
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Simplify \(4\sqrt{81}\): \[ \sqrt{81} = 9. \] Therefore, \[ 4\sqrt{81} = 4 \times 9 = 36. \]
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Calculate \((36)^3\): \[ 36^3 = 36 \times 36 \times 36. \] We will leave this as \(36^3\) for now and proceed to the next part of the equation.
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Write the entire expression: So now we rewrite the expression: \[ 3^{-5} \times (36^3) / 3^{-4}. \]
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Apply the Quotient Property of Exponents: Recall that \(a^m / a^n = a^{m-n}\). Then, \[ \frac{3^{-5}}{3^{-4}} = 3^{-5 - (-4)} = 3^{-5 + 4} = 3^{-1}. \]
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Combine everything: The equation now becomes: \[ 3^{-1} \times 36^3. \]
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Final step: However, we need to go further to equate this to \(3^{-6}\). Let's explore \(36^3\): \[ 36 = 6^2 = (2 \times 3)^2 = 2^2 \times 3^2. \] So, \[ 36^3 = (2^2 \times 3^2)^3 = 2^6 \times 3^6. \] This means, \[ 3^{-1} \times (2^6 \times 3^6) = 2^6 \times 3^{6 - 1} = 2^6 \times 3^5. \]
Now we compare \(2^6 \times 3^5\) with \(3^{-6}\):
- The left-hand side \(2^6 \times 3^5\) cannot equal \(3^{-6}\) because of the \(2^6\) term.
Based on these steps, the equation \(3^{-5} \times (4\sqrt{81})^3 / 3^{-4} = 3^{-6}\) is False. There is an error in calculation in using the quotient property.
Correct response: False. There is an error in calculation in using the quotient property.