Option b: ![9^{-3}](https://tex.z-dn.net/?f=9%5E%7B-3%7D)
Option c: ![3^{-6}](https://tex.z-dn.net/?f=3%5E%7B-6%7D)
Explanation:
The given expression is ![\frac{1}{729}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B729%7D)
To determine from the options which are equivalent to the expression ![\frac{1}{729}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B729%7D)
Option a: ![(-9)^{3}](https://tex.z-dn.net/?f=%28-9%29%5E%7B3%7D)
Simplifying the expression, we have,
![(-9)^{3}=-9\times-9\times-9=-729](https://tex.z-dn.net/?f=%28-9%29%5E%7B3%7D%3D-9%5Ctimes-9%5Ctimes-9%3D-729)
which is not an equivalent expression to ![\frac{1}{729}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B729%7D)
Hence, Option a is not the correct answer.
Option b: ![9^{-3}](https://tex.z-dn.net/?f=9%5E%7B-3%7D)
Simplifying the expression, we have,
![9^{-3}=\frac{1}{9^{3} } =\frac{1}{729}](https://tex.z-dn.net/?f=9%5E%7B-3%7D%3D%5Cfrac%7B1%7D%7B9%5E%7B3%7D%20%7D%20%3D%5Cfrac%7B1%7D%7B729%7D)
which is an equivalent expression to ![\frac{1}{729}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B729%7D)
Hence, Option b is the correct answer.
Option c: ![3^{-6}](https://tex.z-dn.net/?f=3%5E%7B-6%7D)
Simplifying the expression, we have,
![3^{-6}=\frac{1}{3^{6} } =\frac{1}{729}](https://tex.z-dn.net/?f=3%5E%7B-6%7D%3D%5Cfrac%7B1%7D%7B3%5E%7B6%7D%20%7D%20%3D%5Cfrac%7B1%7D%7B729%7D)
which is an equivalent expression to ![\frac{1}{729}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B729%7D)
Hence, Option c is the correct answer.
Option d: ![\left(\frac{1}{9}\right)^{-6}](https://tex.z-dn.net/?f=%5Cleft%28%5Cfrac%7B1%7D%7B9%7D%5Cright%29%5E%7B-6%7D)
Simplifying the expression, we have,
![\left(\frac{1}{9}\right)^{-6}=\frac{1}{\left(\frac{1}{9}\right)^{6}}=\frac{1}{\frac{1}{531441}}=531441](https://tex.z-dn.net/?f=%5Cleft%28%5Cfrac%7B1%7D%7B9%7D%5Cright%29%5E%7B-6%7D%3D%5Cfrac%7B1%7D%7B%5Cleft%28%5Cfrac%7B1%7D%7B9%7D%5Cright%29%5E%7B6%7D%7D%3D%5Cfrac%7B1%7D%7B%5Cfrac%7B1%7D%7B531441%7D%7D%3D531441)
which is not an equivalent expression to ![\frac{1}{729}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B729%7D)
Hence, Option d is not the correct answer.
Option e: ![\left(\frac{1}{3}\right)^{-6}](https://tex.z-dn.net/?f=%5Cleft%28%5Cfrac%7B1%7D%7B3%7D%5Cright%29%5E%7B-6%7D)
Simplifying the expression, we have,
![\left(\frac{1}{3}\right)^{-6}=\frac{1}{\left(\frac{1}{3}\right)^{6}}=\frac{1}{\frac{1}{729}}=729](https://tex.z-dn.net/?f=%5Cleft%28%5Cfrac%7B1%7D%7B3%7D%5Cright%29%5E%7B-6%7D%3D%5Cfrac%7B1%7D%7B%5Cleft%28%5Cfrac%7B1%7D%7B3%7D%5Cright%29%5E%7B6%7D%7D%3D%5Cfrac%7B1%7D%7B%5Cfrac%7B1%7D%7B729%7D%7D%3D729)
which is not an equivalent expression to ![\frac{1}{729}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B729%7D)
Hence, Option e is not the correct answer.
Option f: ![(-3)^{6}](https://tex.z-dn.net/?f=%28-3%29%5E%7B6%7D)
Simplifying the expression, we have,
![(-3)^{6}=-3\times-3\times-3\times-3\times-3\times-3=-729](https://tex.z-dn.net/?f=%28-3%29%5E%7B6%7D%3D-3%5Ctimes-3%5Ctimes-3%5Ctimes-3%5Ctimes-3%5Ctimes-3%3D-729)
which is not an equivalent expression to ![\frac{1}{729}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B729%7D)
Hence, Option f is not the correct answer.