Select all the expressions that are equivalent to 12^5/3^5. Multi-select Choices: (A)1/4^5 (B)4^5 (C)4 (D)4^0

Recognize Simplification Opportunity: First, we need to simplify the expression 12^5/3^5 by recognizing that 12 can be written as a product of 3 and another integer. 12 = 3 * 4, so we can rewrite 12^5 as (3 * 4)^5.Apply Power of a Product Rule: Now, we apply the power of a product rule, which states that (ab)^n = a^n * b^n. So, (3 * 4)^5 = 3^5 * 4^5.Divide and Simplify: We can now simplify the original expression by dividing 3^5 * 4^5 by 3^5. (3^5 * 4^5) / 3^5 = 4^5, because 3^5 / 3^5 = 1.Compare with Choices: Now we compare the simplified expression 4^5 with the given choices to determine which are equivalent. (A) 1/4^5 is not equivalent because it is the reciprocal of 4^5. (B) 4^5 is equivalent because it is the result of our simplification. (C) 4 is not equivalent because it is 4^1, not 4^5. (D) 4^0 is not equivalent because any non-zero number raised to the power of 0 is 1, not 4^5.

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Algebra 2

Simplify rational expressions

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Q. Select all the expressions that are equivalent to

\frac{12^5}{3^5}

\newline

Multi-select Choices:

\newline

(A)

\frac{1}{4^5}

\newline

(B)

4^5

\newline

(C)

4

\newline

(D)

4^0

Recognize Simplification Opportunity: First, we need to simplify the expression $\frac{12^5}{3^5}$ by recognizing that $12$ can be written as a product of $3$ and another integer. $\newline$ $12 = 3 \times 4$ , so we can rewrite $12^5$ as $(3 \times 4)^5$ .
Apply Power of a Product Rule: Now, we apply the power of a product rule, which states that $(ab)^n = a^n \times b^n$ . So, $(3 \times 4)^5 = 3^5 \times 4^5$ .
Divide and Simplify: We can now simplify the original expression by dividing $3^5 \times 4^5$ by $3^5$ . $\newline$ $(3^5 \times 4^5) / 3^5 = 4^5$ , because $3^5 / 3^5 = 1$ .
Compare with Choices: Now we compare the simplified expression $4^5$ with the given choices to determine which are equivalent. $\newline$ (A) $\frac{1}{4^5}$ is not equivalent because it is the reciprocal of $4^5$ . $\newline$ (B) $4^5$ is equivalent because it is the result of our simplification. $\newline$ (C) $4$ is not equivalent because it is $4^1$ , not $4^5$ . $\newline$ (D) $4^0$ is not equivalent because any non-zero number raised to the power of $0$ is $1$ , not $4^5$ .