Answer:
[tex](9)^{-3}[/tex]
[tex](3)^{-6}[/tex]
Step-by-step explanation:
we have
[tex]\frac{1}{729}[/tex]
Decompose the denominator in prime factors
[tex]729=3^6[/tex]
substitute
[tex]\frac{1}{729}=\frac{1}{3^6}=3^{-6}[/tex]
Verify each case
case 1) we have
[tex](-9)^3=-729[/tex]
[tex]-729 \neq \frac{1}{729}[/tex]
therefore
This expression could not have been evaluated by Jayne
case 2) we have
[tex](9)^{-3}=\frac{1}{9^{3}}=\frac{1}{729}[/tex]
[tex]\frac{1}{729}=\frac{1}{729}[/tex]
therefore
This expression could have been evaluated by Jayne
case 3) we have
[tex](3)^{-6}=\frac{1}{3^{6}}=\frac{1}{729}[/tex]
[tex]\frac{1}{729}=\frac{1}{729}[/tex]
therefore
This expression could have been evaluated by Jayne
case 4) we have
[tex](\frac{1}{9})^{-6}=9^{6}=531,441[/tex]
[tex]531,441 \neq \frac{1}{729}[/tex]
therefore
This expression could not have been evaluated by Jayne
case 5) we have
[tex](\frac{1}{3})^{-6}=3^{6}=729[/tex]
[tex]729 \neq \frac{1}{729}[/tex]
therefore
This expression could not have been evaluated by Jayne
case 6) we have
[tex](-3)^6=729[/tex]
[tex]729 \neq \frac{1}{729}[/tex]
therefore
This expression could not have been evaluated by Jayne
