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Which expression is equivalent to $8^5 \times 4^5$ ? $\newline$ Choices: $\newline$ (A) $\frac{1}{32^5}$ $\newline$ (B) $32^{25}$ $\newline$ (C) $32^5$ $\newline$ (D) $32^{10}$

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Identify equivalent expressions involving exponents I

Full solution

Q. Which expression is equivalent to

8^5 \times 4^5

?

\newline

Choices:

\newline

(A)

\frac{1}{32^5}

\newline

(B)

32^{25}

\newline

(C)

32^5

\newline

(D)

32^{10}

Simplify Bases: Simplify the bases. $\newline$ $8 = 2^3$ and $4 = 2^2$ . Rewrite the original expression using these bases. $\newline$ $8^5 \times 4^5 = (2^3)^5 \times (2^2)^5$
Apply Power Rule: Apply the power of a power rule. $\newline$ $(2^3)^5 = 2^{(3\times5)}$ and $(2^2)^5 = 2^{(2\times5)}$ $\newline$ $2^{(3\times5)} \times 2^{(2\times5)} = 2^{15} \times 2^{10}$
Add Exponents: Add the exponents since the bases are the same. $\newline$ $2^{15} \times 2^{10} = 2^{(15+10)} = 2^{25}$
Recognize Error: Recognize the error in exponent calculation. $\newline$ $2^{25}$ is not an option in the choices, recheck calculations. $\newline$ $2^{15} \times 2^{10}$ should be $2^{(15+10)} = 2^{25}$ , but we need to express it in terms of $32$ . $\newline$ $32 = 2^{5}$ , so $2^{25} = (2^{5})^{5} = 32^{5}$

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