Select all the expressions that are equivalent to 6^3 / 3^3.
2 · 2²
2
2^6 / 2²
2³
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Select all the expressions that are equivalent to 6^3 / 3^3.
2 · 2²
2
2^6 / 2²
2³
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 (