Select all the expressions that are equivalent to $\frac{6^5}{2^5}$ . $\newline$ Multi-select Choices: $\newline$ (A) $\frac{1}{3^5}$ $\newline$ (B) $3$ $\newline$ (C) $\frac{1}{3^{-5}}$ $\newline$ (D) $3^5$

Full solution

Q. Select all the expressions that are equivalent to

\frac{6^5}{2^5}

\newline

Multi-select Choices:

\newline

(A)

\frac{1}{3^5}

\newline

(B)

3

\newline

(C)

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

\newline

(D)

3^5

Break down base $6$ : Simplify the expression $6^5/2^5$ by breaking down the base $6$ into its prime factors. $\newline$ $6$ can be expressed as $2 \times 3$ , so $6^5$ can be written as $(2 \times 3)^5$ .
Apply power to factors: Apply the power to each factor inside the parentheses. $\newline$ $(2 \times 3)^5 = 2^5 \times 3^5$
Divide by $2^5$ : Divide $2^5 \times 3^5$ by $2^5$ . $\newline$ Since we have $2^5$ in both the numerator and the denominator, they cancel each other out. $\newline$ $2^5 \times 3^5 / 2^5 = 3^5$
Compare with choices: Compare the result with the given choices. $\newline$ The expression $\frac{6^5}{2^5}$ simplifies to $3^5$ . Now we need to check which of the given choices are equivalent to $3^5$ .
Evaluate each choice: Evaluate each choice to see if it is equivalent to $3^5$ .
(A) $\frac{1}{3^5}$ is not equivalent because it is the reciprocal of $3^5$ .
(B) $3$ is not equivalent because it is $3^1$ , not $3^5$ .
(C) $\frac{1}{3^{-5}}$ is equivalent because the negative exponent means the reciprocal, so it simplifies to $3^5$ .
(D) $3^5$ is clearly equivalent because it is the same expression.