Option b: [tex]9^{-3}[/tex]
Option c: [tex]3^{-6}[/tex]
Explanation:
The given expression is [tex]\frac{1}{729}[/tex]
To determine from the options which are equivalent to the expression [tex]\frac{1}{729}[/tex]
Option a: [tex](-9)^{3}[/tex]
Simplifying the expression, we have,
[tex](-9)^{3}=-9\times-9\times-9=-729[/tex]
which is not an equivalent expression to [tex]\frac{1}{729}[/tex]
Hence, Option a is not the correct answer.
Option b: [tex]9^{-3}[/tex]
Simplifying the expression, we have,
[tex]9^{-3}=\frac{1}{9^{3} } =\frac{1}{729}[/tex]
which is an equivalent expression to [tex]\frac{1}{729}[/tex]
Hence, Option b is the correct answer.
Option c: [tex]3^{-6}[/tex]
Simplifying the expression, we have,
[tex]3^{-6}=\frac{1}{3^{6} } =\frac{1}{729}[/tex]
which is an equivalent expression to [tex]\frac{1}{729}[/tex]
Hence, Option c is the correct answer.
Option d: [tex]\left(\frac{1}{9}\right)^{-6}[/tex]
Simplifying the expression, we have,
[tex]\left(\frac{1}{9}\right)^{-6}=\frac{1}{\left(\frac{1}{9}\right)^{6}}=\frac{1}{\frac{1}{531441}}=531441[/tex]
which is not an equivalent expression to [tex]\frac{1}{729}[/tex]
Hence, Option d is not the correct answer.
Option e: [tex]\left(\frac{1}{3}\right)^{-6}[/tex]
Simplifying the expression, we have,
[tex]\left(\frac{1}{3}\right)^{-6}=\frac{1}{\left(\frac{1}{3}\right)^{6}}=\frac{1}{\frac{1}{729}}=729[/tex]
which is not an equivalent expression to [tex]\frac{1}{729}[/tex]
Hence, Option e is not the correct answer.
Option f: [tex](-3)^{6}[/tex]
Simplifying the expression, we have,
[tex](-3)^{6}=-3\times-3\times-3\times-3\times-3\times-3=-729[/tex]
which is not an equivalent expression to [tex]\frac{1}{729}[/tex]
Hence, Option f is not the correct answer.
