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Which of the following expressions is equivalent to
root(4)(32^(16)) ?
Choose 1 answer:
(A)
2^(20)
(B)
2^(625)
(C)
32^(2)
(D)
32^(12)

Which of the following expressions is equivalent to $\sqrt[4]{32^{16}}$ ? $\newline$ Choose $1$ answer: $\newline$ (A) $2^{20}$ $\newline$ (B) $2^{625}$ $\newline$ (C) $32^{2}$ $\newline$ (D) $32^{12}$

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Identify equivalent exponential expressions II

Full solution

Q. Which of the following expressions is equivalent to

\sqrt[4]{32^{16}}

?

\newline

Choose

1

answer:

\newline

(A)

2^{20}

\newline

(B)

2^{625}

\newline

(C)

32^{2}

\newline

(D)

32^{12}

Express as Power of $2$ : First, let's express $32$ as a power of $2$ since $32$ is $2^5$ . $\newline$ $\sqrt[4]{32^{16}} = \sqrt[4]{(2^5)^{16}}$
Apply Power Rule: Now, apply the power rule $(a^{m})^{n} = a^{m*n}$ . $\newline$ $\sqrt[4]{(2^{5})^{16}} = \sqrt[4]{2^{5*16}}$
Multiply Exponents: Multiply the exponents inside the root. $\sqrt[4]{2^{5\times16}} = \sqrt[4]{2^{80}}$
Use Fourth Root Rule: The fourth root of a number is the same as raising that number to the $\frac{1}{4}$ power. $\newline$ $\sqrt[4]{2^{80}} = (2^{80})^{\frac{1}{4}}$
Apply Power Rule Again: Apply the power rule again $(a^m)^n = a^{m*n}$ . $(2^{80})^{1/4} = 2^{80*1/4}$
Simplify Exponents: Multiply the exponents to simplify. $2^{80\times\frac{1}{4}} = 2^{\frac{80}{4}}$
Divide to Find Exponent: Divide $80$ by $4$ to find the exponent. $\newline$ $2^{(80/4)} = 2^{20}$

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