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Question Choose the correct expression equivalent to 22⋅232^{2} \cdot 2^{3}, 2521\frac{2^{5}}{2^{1}}, and (22)3\left(2^{2}\right)^{3}.

Studdy Solution

STEP 1

Assumptions
1. We are given three expressions: 22⋅232^{2} \cdot 2^{3}, 2521\frac{2^{5}}{2^{1}}, and (22)3\left(2^{2}\right)^{3}.
2. We need to determine which of these expressions is equivalent to 242^{4}, 252^{5}, and 262^{6}.
3. The laws of exponents will be used to simplify the expressions.

STEP 2

Simplify the first expression using the law of exponents which states that when multiplying powers with the same base, you add the exponents.
22⋅23=22+32^{2} \cdot 2^{3} = 2^{2+3}

STEP 3

Calculate the exponent for the first expression.
22+3=252^{2+3} = 2^{5}

STEP 4

Mark the equivalent expression for 252^{5}.
\begin{tabular}{|l|c|c|c|} \hline & 242^{4} & 252^{5} & 262^{6} \\ \hline 22⋅232^{2} \cdot 2^{3} & □\square & ✓\checkmark & □\square \\ \hline2521\frac{2^{5}}{2^{1}} & □\square & □\square & □\square \\ \hline(22)3\left(2^{2}\right)^{3} & □\square & □\square & □\square \\ \hline \end{tabular}

STEP 5

Simplify the second expression using the law of exponents which states that when dividing powers with the same base, you subtract the exponents.
2521=25−1\frac{2^{5}}{2^{1}} = 2^{5-1}

STEP 6

Calculate the exponent for the second expression.
25−1=242^{5-1} = 2^{4}

STEP 7

Mark the equivalent expression for 242^{4}.
\begin{tabular}{|l|c|c|c|} \hline & 242^{4} & 252^{5} & 262^{6} \\ \hline 22⋅232^{2} \cdot 2^{3} & □\square & ✓\checkmark & □\square \\ \hline2521\frac{2^{5}}{2^{1}} & ✓\checkmark & □\square & □\square \\ \hline(22)3\left(2^{2}\right)^{3} & □\square & □\square & □\square \\ \hline \end{tabular}

STEP 8

Simplify the third expression using the law of exponents which states that when raising a power to a power, you multiply the exponents.
(22)3=22⋅3\left(2^{2}\right)^{3} = 2^{2 \cdot 3}

STEP 9

Calculate the exponent for the third expression.
22⋅3=262^{2 \cdot 3} = 2^{6}

STEP 10

Mark the equivalent expression for 262^{6}.
\begin{tabular}{|l|c|c|c|} \hline & 242^{4} & 252^{5} & 262^{6} \\ \hline 22⋅232^{2} \cdot 2^{3} & □\square & ✓\checkmark & □\square \\ \hline2521\frac{2^{5}}{2^{1}} & ✓\checkmark & □\square & □\square \\ \hline(22)3\left(2^{2}\right)^{3} & □\square & □\square & ✓\checkmark \\ \hline \end{tabular}
The correct equivalent expressions are:
- 22⋅232^{2} \cdot 2^{3} is equivalent to 252^{5} - 2521\frac{2^{5}}{2^{1}} is equivalent to 242^{4} - (22)3\left(2^{2}\right)^{3} is equivalent to 262^{6}

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