Select all the expressions that are equivalent to 6^3 / 3^3

To simplify the expression ( frac{6^3}{3^3} ), we can first rewrite ( 6 ) in terms of its prime factors. ( 6 = 2 cdot 3 ), so:

[

6^3 = (2 cdot 3)^3 = 2^3 cdot 3^3

]

Now substituting in our original expression:

[

frac{6^3}{3^3} = frac{2^3 cdot 3^3}{3^3}

]

The ( 3^3 ) cancels out, leaving us with:

[

2^3

]

Now, let’s check which expressions are equivalent to ( 2^3 ):

1. ( 2 cdot 2^2 ): This simplifies to ( 2^1 cdot 2^2 = 2^{1+2} = 2^3 ) → Equivalent
2. ( 2 ): This is ( 2^1 ) → Not equivalent
3. ( frac{2^6}{2^2} ): This simplifies to ( 2^{6-2} = 2^4 ) → Not equivalent
4. ( 2^3 ): This is equal to our simplified expression → Equivalent

Thus, the expressions equivalent to (