Choose the four expressions that are equivalent to 3⁻⁴.

To find the expressions equivalent to ( 3^{-4} ), we first recall that ( a^{-n} = frac{1}{a^n} ). This means that ( 3^{-4} = frac{1}{3^4} = frac{1}{81} ).

Now, let’s evaluate each expression:

1) ( frac{1}{81} ): This is ( 3^{-4} ), so it’s equivalent.

2) ( frac{1}{3^4} ): This is also ( 3^{-4} ), so it’s equivalent.

3) ( 3^{1} times 3^{-4} ): This simplifies to ( 3^{-3} ), which is not equivalent to ( 3^{-4} ).

4) ( (3^{-7})^3 ): This simplifies to ( 3^{-21} ), which is not equivalent to ( 3^{-4} ).

5) ( (1/3)^4 ): This is ( 3^{-4} ), so it’s equivalent.

6) ( frac{3^2}{3^6} ): This simplifies to ( 3^{-4} ), so it’s equivalent.

So, the four expressions that are equivalent to ( 3^{-4} ) are:

1) ( frac{1}{81} )

2) ( frac{1}{3