Question

Jayne evaluated an expression that has a value of StartFraction 1 Over 729 EndFraction. Which expression could Jayne have evaluated? Check all that apply.
(negative 9) cube
9 Superscript negative 3
3 Superscript negative 6
(StartFraction 1 Over 9 EndFraction) Superscript negative 6
(one-third) Superscript negative 6
(Negative 3) Superscript negative 6

Answer 1

To determine which expressions Jayne could have evaluated to get the value of \( \frac{1}{729} \), we can calculate the value of each expression and see if it simplifies to \( \frac{1}{729} \).

1. **Expression: \( (-9)^3 \)**
\[
(-9)^3 = -729 \quad \text{(not equal to \( \frac{1}{729} \))}
\]

2. **Expression: \( 9^{-3} \)**
\[
9^{-3} = \frac{1}{9^3} = \frac{1}{729} \quad \text{(this is a match)}
\]

3. **Expression: \( 3^{-6} \)**
\[
3^{-6} = \frac{1}{3^6} = \frac{1}{729} \quad \text{(this is a match)}
\]

4. **Expression: \( \left(\frac{1}{9}\right)^{-6} \)**
\[
\left(\frac{1}{9}\right)^{-6} = 9^6 = (3^2)^6 = 3^{12} \quad \text{(not equal to \( \frac{1}{729} \))}
\]

5. **Expression: \( \left(\frac{1}{3}\right)^{-6} \)**
\[
\left(\frac{1}{3}\right)^{-6} = 3^6 = 729 \quad \text{(not equal to \( \frac{1}{729} \))}
\]

6. **Expression: \( (-3)^{-6} \)**
\[
(-3)^{-6} = \frac{1}{(-3)^6} = \frac{1}{729} \quad \text{(this is a match)}
\]

### Summary of Matching Expressions:
- \( 9^{-3} \)
- \( 3^{-6} \)
- \( (-3)^{-6} \)

Thus, the expressions that Jayne could have evaluated to get \( \frac{1}{729} \) are:
- \( 9^{-3} \)
- \( 3^{-6} \)
- \( (-3)^{-6} \)