Question

is the equation (3^(-5)*(\root(4)(81))^(3))/(3^(-4))=3^(-6) true or false. select the responses that correctly answers the questions and provides the appropriate justification

true, but the equation should be simplified using the negative exponent rule
false. there is an error in applying the product property
false. there is an error in calculation using the quotient property
true. al rules have been applied correctly

Answer 1

choose a, b, c, or d

Answer 2

To determine whether the equation

\[
\frac{3^{-5} \cdot (\sqrt[4]{81})^3}{3^{-4}} = 3^{-6}
\]

is true or false, let's simplify both sides step by step.

1. **Simplify \(\sqrt[4]{81}\)**:
\[
\sqrt[4]{81} = \sqrt[4]{3^4} = 3^{4/4} = 3^1 = 3
\]

2. **Now substitute this back into the equation**:
\[
\frac{3^{-5} \cdot (3)^3}{3^{-4}} = \frac{3^{-5} \cdot 3^3}{3^{-4}}
\]

3. **Using the product property of exponents**:
\[
3^{-5} \cdot 3^3 = 3^{-5 + 3} = 3^{-2}
\]

4. **Thus, we now have**:
\[
\frac{3^{-2}}{3^{-4}}
\]

5. **Using the quotient property of exponents**:
\[
\frac{3^{-2}}{3^{-4}} = 3^{-2 - (-4)} = 3^{-2 + 4} = 3^{2}
\]

6. **So the left side simplifies to \(3^2\)**.

7. **Now compare this to the right side**:
On the right side, we have \(3^{-6}\).

Since \(3^2 \neq 3^{-6}\), the original equation is **false**.

Now let's choose the correct justification based on our analysis:

- **false. there is an error in calculation using the quotient property** is the most appropriate response because the simplification led to \(3^2\) instead of \(3^{-6}\).

Therefore, the correct response is:
**false. there is an error in calculation using the quotient property.**

Answer 3

Based on the analysis, the correct choice is:

**c. false. there is an error in calculation using the quotient property.**