To determine which expressions could Jayne have evaluated to get the value [tex]\(\frac{1}{729}\)[/tex], we need to check each given expression. Let's evaluate and verify each expression one by one:
1. Expression: [tex]\((-9)^3\)[/tex]
[tex]\[
(-9)^3 = -9 \times -9 \times -9 = -729
\][/tex]
This does not equal [tex]\(\frac{1}{729}\)[/tex].
2. Expression: [tex]\(9^{-3}\)[/tex]
[tex]\[
9^{-3} = \left(\frac{1}{9}\right)^3 = \frac{1}{9^3} = \frac{1}{729}
\][/tex]
This equals [tex]\(\frac{1}{729}\)[/tex].
3. Expression: [tex]\(3^{-6}\)[/tex]
[tex]\[
3^{-6} = \left(\frac{1}{3}\right)^6 = \frac{1}{3^6} = \frac{1}{729}
\][/tex]
This equals [tex]\(\frac{1}{729}\)[/tex].
4. Expression: [tex]\(\left(\frac{1}{9}\right)^{-6}\)[/tex]
[tex]\[
\left(\frac{1}{9}\right)^{-6} = 9^6
\][/tex]
[tex]\(9^6\)[/tex] is not equal to [tex]\(\frac{1}{729}\)[/tex].
5. Expression: [tex]\(\left(\frac{1}{3}\right)^{-6}\)[/tex]
[tex]\[
\left(\frac{1}{3}\right)^{-6} = 3^6
\][/tex]
[tex]\(3^6\)[/tex] is not equal to [tex]\(\frac{1}{729}\)[/tex].
6. Expression: [tex]\((-3)^6\)[/tex]
[tex]\[
(-3)^6 = (-3) \times (-3) \times (-3) \times (-3) \times (-3) \times (-3) = 729
\][/tex]
This does not equal [tex]\(\frac{1}{729}\)[/tex].
Therefore, the expressions that Jayne could have evaluated to get the value [tex]\(\frac{1}{729}\)[/tex] are:
[tex]\(9^{-3}\)[/tex] and [tex]\(3^{-6}\)[/tex].
Checked Expressions:
- [tex]\((-9)^3\)[/tex] does not equal [tex]\(\frac{1}{729}\)[/tex]
- [tex]\(9^{-3}\)[/tex] equals [tex]\(\frac{1}{729}\)[/tex]
- [tex]\(3^{-6}\)[/tex] equals [tex]\(\frac{1}{729}\)[/tex]
- [tex]\(\left(\frac{1}{9}\right)^{-6}\)[/tex] does not equal [tex]\(\frac{1}{729}\)[/tex]
- [tex]\(\left(\frac{1}{3}\right)^{-6}\)[/tex] does not equal [tex]\(\frac{1}{729}\)[/tex]
- [tex]\((-3)^6\)[/tex] does not equal [tex]\(\frac{1}{729}\)[/tex]
The valid expressions are:
[tex]\(\boxed{9^{-3}}\)[/tex] and [tex]\(\boxed{3^{-6}}\)[/tex]
