One of the numerical expressions below is equivalent to $2^{2} \cdot 32^{\frac{1}{2}}$ . Which one? $\newline$ A $32 \sqrt{2}$ $\newline$ B $128^{\frac{1}{2}}$ $\newline$ C $16 \sqrt{2}$ $\newline$ D $64^{\frac{5}{2}}$

View step-by-step help

Home

Math Problems

Algebra 2

Multiplication with rational exponents

Full solution

Q. One of the numerical expressions below is equivalent to

2^{2} \cdot 32^{\frac{1}{2}}

. Which one?

\newline

32 \sqrt{2}

\newline

128^{\frac{1}{2}}

\newline

16 \sqrt{2}

\newline

64^{\frac{5}{2}}

Simplify $2^2$ : First, let's simplify $2^{2}$ which is $2*2$ . $\newline$ $2^{2} = 4$
Find square root of $32$ : Now, let's find the square root of $32$ , which is the same as $32^{1/2}$ . $32^{1/2} = \sqrt{32}$
Simplify $\sqrt{32}$ : We know that $32$ is $2^5$ , so $\sqrt{32}$ is the same as $(2^5)^{1/2}$ . $\newline$ $(2^5)^{1/2}$ = $2^{5/2}$
Multiply simplified terms: Now, multiply the simplified terms $4$ and $2^{5/2}$ together. $4 \times 2^{5/2} = 2^{2} \times 2^{5/2}$
Add exponents: When multiplying powers with the same base, we add the exponents. $2^{2} \times 2^{\frac{5}{2}} = 2^{2 + \frac{5}{2}}$
Calculate $2^{9/2}$ : Add the exponents: $2 + \frac{5}{2}$ is the same as $\frac{4}{2} + \frac{5}{2}$ . $\newline$ $2 + \frac{5}{2} = \frac{4}{2} + \frac{5}{2} = \frac{9}{2}$
Check answer choices: Now we have $2^{\frac{9}{2}}$ , which is the simplified form of the original expression. $\newline$ $2^{\frac{9}{2}}$
Check answer choices: Now we have $2^{\frac{9}{2}}$ , which is the simplified form of the original expression. $\newline$ $2^{\frac{9}{2}}$ Let's check the answer choices to see which one matches $2^{\frac{9}{2}}$ . $\newline$ A) $32\sqrt{2}$ is not in the form of a power of $2$ . $\newline$ B) $128^{\frac{1}{2}}$ is the same as $2^{\frac{7}{2}}$ , which is not $2^{\frac{9}{2}}$ . $\newline$ C) $16\sqrt{2}$ is not in the form of a power of $2$ . $\newline$ D) $2^{\frac{9}{2}}$ $0$ is the same as $2^{\frac{9}{2}}$ $1$ which is $2^{\frac{9}{2}}$ $2$ , which is not $2^{\frac{9}{2}}$ .
Check answer choices: Now we have $2^{\frac{9}{2}}$ , which is the simplified form of the original expression. $\newline$ $2^{\frac{9}{2}}$ Let's check the answer choices to see which one matches $2^{\frac{9}{2}}$ . $\newline$ A) $32\sqrt{2}$ is not in the form of a power of $2$ . $\newline$ B) $128^{\frac{1}{2}}$ is the same as $2^{\frac{7}{2}}$ , which is not $2^{\frac{9}{2}}$ . $\newline$ C) $16\sqrt{2}$ is not in the form of a power of $2$ . $\newline$ D) $2^{\frac{9}{2}}$ $0$ is the same as $2^{\frac{9}{2}}$ $1$ which is $2^{\frac{9}{2}}$ $2$ , which is not $2^{\frac{9}{2}}$ .None of the answer choices match $2^{\frac{9}{2}}$ . There must be a mistake in the answer choices or the question itself.