Select all the expressions that are equivalent to $3^6 \times 3^{-9}$ . $\newline$ Multi-select Choices: $\newline$ (A) $\frac{1}{3^3}$ $\newline$ (B) $\frac{1}{3^{-54}}$ $\newline$ (C) $\frac{1}{3^{-3}}$ $\newline$ (D) $3^{-54}$

Full solution

Q. Select all the expressions that are equivalent to

3^6 \times 3^{-9}

\newline

Multi-select Choices:

\newline

(A)

\frac{1}{3^3}

\newline

(B)

\frac{1}{3^{-54}}

\newline

(C)

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

\newline

(D)

3^{-54}

Simplify Exponent Expression: Simplify the expression $3^6 \times 3^{-9}$ . When multiplying powers with the same base, we add the exponents. $3^6 \times 3^{-9} = 3^{6 + (-9)} = 3^{-3}$
Convert Negative Exponent: Convert the negative exponent to a positive exponent by writing it as a fraction. $3^{-3}$ is equivalent to $\frac{1}{3^3}$ because any number with a negative exponent can be written as the reciprocal with a positive exponent.
Compare with Choices: Compare the simplified expression to the multi-select choices. $\newline$ The simplified expression is $\frac{1}{3^3}$ , which matches choice $(A)$ .
Check Other Equivalences: Check the other choices for equivalence. $\newline$ (B) $\frac{1}{3^{-54}}$ is not equivalent because the exponent $-54$ does not match $-3$ . $\newline$ (C) $\frac{1}{3^{-3}}$ is equivalent because it is the reciprocal of $3^{-3}$ , which is the same as $3^{-3}$ . $\newline$ (D) $3^{-54}$ is not equivalent because the exponent $-54$ does not match $-3$ .