Question

Is the equation (3 ^ - 5 * (root(81, 4)) ^ 3)/(3 ^ - 4) = 3 ^ - 6 true or false? Select the response that correctly answers the question and provides the appropriate justification. False. There is an error in calculation using the Quotient Property. True. All rules have been applied correctly. False. There is an error in applying the Product Property. True, but the equation should be simplified using the Negative Exponent Rule.

Answer 1

To evaluate the equation \((3^{-5} \cdot (\sqrt[4]{81})^3) / (3^{-4}) = 3^{-6}\), let's simplify the left side step by step.

1. **Simplify \(\sqrt[4]{81}\)**:
\[
81 = 3^4 \implies \sqrt[4]{81} = \sqrt[4]{3^4} = 3
\]
So, \((\sqrt[4]{81})^3 = 3^3 = 27\).

2. **Substitute into the equation**:
\[
3^{-5} \cdot (3^3) / (3^{-4}) = 3^{-5} \cdot 27 / 3^{-4}
\]

3. **Convert 27 to a power of 3**:
\[
27 = 3^3
\]
So the expression becomes:
\[
3^{-5} \cdot 3^3 / 3^{-4}
\]

4. **Combine using the Product Property**:
\[
3^{-5 + 3} / 3^{-4} = 3^{-2} / 3^{-4}
\]

5. **Apply the Quotient Property**:
\[
3^{-2 - (-4)} = 3^{-2 + 4} = 3^2
\]

The left side simplifies to:
\[
3^2
\]

Now compare that to the right side \(3^{-6}\):
Since \(3^2 \neq 3^{-6}\), the original equation is false.

**Final Answer**: **False. There is an error in calculation using the Quotient Property.**