Solved on Dec 15, 2023

Choose the correct expression equivalent to $2^{2} \cdot 2^{3}$ , $\frac{2^{5}}{2^{1}}$ , and $\left(2^{2}\right)^{3}$ .

STEP 1

Assumptions
1. We are given three expressions: $2^{2} \cdot 2^{3}$ , $\frac{2^{5}}{2^{1}}$ , and $\left(2^{2}\right)^{3}$ .
2. We need to determine which of these expressions is equivalent to $2^{4}$ , $2^{5}$ , and $2^{6}$ .
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.
$2^{2} \cdot 2^{3} = 2^{2+3}$

STEP 3

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

STEP 4

Mark the equivalent expression for $2^{5}$ .
\begin{tabular}{|l|c|c|c|} \hline & $2^{4}$ & $2^{5}$ & $2^{6}$ \\ \hline $2^{2} \cdot 2^{3}$ & $\square$ & $\checkmark$ & $\square$ \\ \hline $\frac{2^{5}}{2^{1}}$ & $\square$ & $\square$ & $\square$ \\ \hline $\left(2^{2}\right)^{3}$ & $\square$ & $\square$ & $\square$ \\ \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.
$\frac{2^{5}}{2^{1}} = 2^{5-1}$

STEP 6

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

STEP 7

Mark the equivalent expression for $2^{4}$ .
\begin{tabular}{|l|c|c|c|} \hline & $2^{4}$ & $2^{5}$ & $2^{6}$ \\ \hline $2^{2} \cdot 2^{3}$ & $\square$ & $\checkmark$ & $\square$ \\ \hline $\frac{2^{5}}{2^{1}}$ & $\checkmark$ & $\square$ & $\square$ \\ \hline $\left(2^{2}\right)^{3}$ & $\square$ & $\square$ & $\square$ \\ \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.
$\left(2^{2}\right)^{3} = 2^{2 \cdot 3}$

STEP 9

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

STEP 10

Mark the equivalent expression for $2^{6}$ .
\begin{tabular}{|l|c|c|c|} \hline & $2^{4}$ & $2^{5}$ & $2^{6}$ \\ \hline $2^{2} \cdot 2^{3}$ & $\square$ & $\checkmark$ & $\square$ \\ \hline $\frac{2^{5}}{2^{1}}$ & $\checkmark$ & $\square$ & $\square$ \\ \hline $\left(2^{2}\right)^{3}$ & $\square$ & $\square$ & $\checkmark$ \\ \hline \end{tabular}
The correct equivalent expressions are:
- $2^{2} \cdot 2^{3}$ is equivalent to $2^{5}$ - $\frac{2^{5}}{2^{1}}$ is equivalent to $2^{4}$ - $\left(2^{2}\right)^{3}$ is equivalent to $2^{6}$

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Choose the correct expression equivalent to $2^{2} \cdot 2^{3}$ , $\frac{2^{5}}{2^{1}}$ , and $\left(2^{2}\right)^{3}$ .

STEP 1

Assumptions
1. We are given three expressions: $2^{2} \cdot 2^{3}$ , $\frac{2^{5}}{2^{1}}$ , and $\left(2^{2}\right)^{3}$ .
2. We need to determine which of these expressions is equivalent to $2^{4}$ , $2^{5}$ , and $2^{6}$ .
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.
$2^{2} \cdot 2^{3} = 2^{2+3}$

STEP 3

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

STEP 4

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.
$\frac{2^{5}}{2^{1}} = 2^{5-1}$

STEP 6

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

STEP 7

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.
$\left(2^{2}\right)^{3} = 2^{2 \cdot 3}$

STEP 9

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

STEP 10

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Choose the correct expression equivalent to 22⋅232^{2} \cdot 2^{3}22⋅23, 2521\frac{2^{5}}{2^{1}}2125​, and (22)3\left(2^{2}\right)^{3}(22)3.

STEP 1

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}22⋅23=22+3

STEP 3

Calculate the exponent for the first expression.22+3=252^{2+3} = 2^{5}22+3=25

STEP 4

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}2125​=25−1

STEP 6

Calculate the exponent for the second expression.25−1=242^{5-1} = 2^{4}25−1=24

STEP 7

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}(22)3=22⋅3

STEP 9

Calculate the exponent for the third expression.22⋅3=262^{2 \cdot 3} = 2^{6}22⋅3=26

STEP 10

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Choose the correct expression equivalent to $2^{2} \cdot 2^{3}$ , $\frac{2^{5}}{2^{1}}$ , and $\left(2^{2}\right)^{3}$ .

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

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

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

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

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

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