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Select all the expressions that are equivalent to $(3^{-5})^{-1}$ . $\newline$ Multi-select Choices: $\newline$ (A) $\frac{1}{3^5}$ $\newline$ (B) $3^{-6}$ $\newline$ (C) $\frac{1}{3^{-6}}$ $\newline$ (D) $3^5$

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Simplify exponential expressions using exponent rules

Full solution

Q. Select all the expressions that are equivalent to

(3^{-5})^{-1}

.

\newline

Multi-select Choices:

\newline

(A)

\frac{1}{3^5}

\newline

(B)

3^{-6}

\newline

(C)

\frac{1}{3^{-6}}

\newline

(D)

3^5

Exponent Rule Explanation: Understand the exponent rule for powers raised to powers. $\newline$ When we have an expression of the form $(a^b)^c$ , we multiply the exponents $b$ and $c$ . $\newline$ $(3^{-5})^{-1} = 3^{(-5 \times -1)} = 3^5$
Simplify the Expression: Compare the result from Step $1$ with the given choices. $\newline$ We found that $(3^{-5})^{-1}$ simplifies to $3^5$ . Now we need to determine which of the given choices are equivalent to $3^5$ . $\newline$ (A) $\frac{1}{3^5}$ is the reciprocal of $3^5$ , so it is not equivalent. $\newline$ (B) $3^{-6}$ is $3$ raised to the negative sixth power, which is not the same as $3^5$ . $\newline$ (C) $\frac{1}{3^{-6}}$ is the reciprocal of $3^{-6}$ , which simplifies to $3^5$ $0$ , not $3^5$ . $\newline$ (D) $3^5$ is exactly what we found in Step $1$ , so it is equivalent.

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