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Select all the expressions that are equivalent to $5^2 \times 5^{-5}$ . $\newline$ Multi-select Choices: $\newline$ (A) $\frac{1}{5^{-10}}$ $\newline$ (B) $5^{-3}$ $\newline$ (C) $5^{-10}$ $\newline$ (D) $\frac{1}{5^{-3}}$

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Simplify exponential expressions using exponent rules

Full solution

Q. Select all the expressions that are equivalent to

5^2 \times 5^{-5}

.

\newline

Multi-select Choices:

\newline

(A)

\frac{1}{5^{-10}}

\newline

(B)

5^{-3}

\newline

(C)

5^{-10}

\newline

(D)

\frac{1}{5^{-3}}

Simplify Expression: Simplify the given expression $5^2 \times 5^{-5}$ . $\newline$ When multiplying powers with the same base, we add the exponents. $\newline$ $5^2 \times 5^{-5} = 5^{2 + (-5)} = 5^{-3}$
Compare to Choices: Compare the simplified expression to the choices. $\newline$ We have found that $5^2 \times 5^{-5}$ simplifies to $5^{-3}$ . Now we need to check which of the given choices are equivalent to $5^{-3}$ . $\newline$ (A) $\frac{1}{5^{-10}}$ is equivalent to $5^{10}$ , which is not equal to $5^{-3}$ .
Check Choice (A): Check choice (B). $\newline$ (B) $5^{-3}$ is exactly the expression we found in Step $1$ , so it is equivalent to $5^2 \times 5^{-5}.$
Check Choice (B): Check choice (C). $\newline$ (C) $5^{-10}$ is not equivalent to $5^{-3}$ because the exponents are different.
Check Choice (C): Check choice (D). $\newline$ (D) $\frac{1}{5^{-3}}$ is equivalent to $5^3$ , which is not equal to $5^{-3}$ .

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