Select all the expressions that are equivalent to $3^{-8} \times 3^{-1}$ . $\newline$ Multi-select Choices: $\newline$ (A) $\frac{1}{3^8}$ $\newline$ (B) $3^{-9}$ $\newline$ (C) $3^8$ $\newline$ (D) $\frac{1}{3^{-9}}$

Full solution

Q. Select all the expressions that are equivalent to

3^{-8} \times 3^{-1}

\newline

Multi-select Choices:

\newline

(A)

\frac{1}{3^8}

\newline

(B)

3^{-9}

\newline

(C)

3^8

\newline

(D)

\frac{1}{3^{-9}}

Understand the problem: Understand the problem. $\newline$ We need to find which expressions are equivalent to the multiplication of two powers of $3$ with negative exponents: $3^{-8}$ and $3^{-1}$ .
Apply exponent rule: Apply the exponent rule for multiplication. $\newline$ When multiplying powers with the same base, we add the exponents. $\newline$ $3^{-8} \times 3^{-1} = 3^{(-8 + (-1))} = 3^{-9}$
Compare with choices: Compare the result with the choices. $\newline$ We have found that $3^{-8} \times 3^{-1}$ simplifies to $3^{-9}$ . Now we need to compare this result with the given choices to determine which are equivalent. $\newline$ (A) $\frac{1}{3^8}$ is not equivalent because it represents $3^{-8}$ , not $3^{-9}$ . $\newline$ (B) $3^{-9}$ is equivalent because it is the result we obtained. $\newline$ (C) $3^8$ is not equivalent because it represents the positive exponent, not the negative. $\newline$ (D) $\frac{1}{3^{-9}}$ is equivalent because it represents the reciprocal of $3^{-9}$ , which is the same as $3^{-9}$ itself.