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)

Express as Power of 2: First, let's express 32 as a power of 2 since 32 is 2^5. root(4)(32^(16)) = root(4)((2^5)^(16))Apply Power Rule: Now, apply the power rule (a^(m))^n = a^(m*n). root(4)((2^5)^(16)) = root(4)(2^(5*16))Multiply Exponents: Multiply the exponents inside the root. root(4)(2^(5*16)) = root(4)(2^(80))Use Fourth Root Rule: The fourth root of a number is the same as raising that number to the 1/4 power. root(4)(2^(80)) = (2^(80))^(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*1/4) = 2^(80/4)Divide to Find Exponent: Divide 80 by 4 to find the exponent. 2^(80/4) = 2^(20)

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Math Problems

Algebra 1

Identify equivalent exponential expressions II

Full solution

Q. Which of the following expressions is equivalent to

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

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Choose

1

answer:

\newline

(A)

2^{20}

\newline

(B)

2^{625}

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(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. $\newline$ $2^{(80/4)} = 2^{20}$