Which expressions are equivalent to $\newline$ $6 \times 6 \times 6 \times 6 \times 6$ ? $\newline$ Choose $2$ answers: $\newline$ A) $(6^{2})^{3}$ $\newline$ B) $2^{5} \times 3^{5}$ $\newline$ C) $\frac{6^{6}}{6^{1}}$ $\newline$ D) $3^{2} \times 2^{3}$

Full solution

Q. Which expressions are equivalent to

\newline

6 \times 6 \times 6 \times 6 \times 6

\newline

Choose

2

answers:

\newline

(6^{2})^{3}

\newline

2^{5} \times 3^{5}

\newline

\frac{6^{6}}{6^{1}}

\newline

3^{2} \times 2^{3}

Understand original expression: Understand the original expression. $\newline$ The original expression is $6*6*6*6*6$ , which is $6$ raised to the power of $5$ , or $6^5$ .
Evaluate option A: Evaluate option A. $\newline$ Option A is $(6^2)^3$ . According to the rule of exponents, $(a^m)^n = a^{m*n}$ . So, $(6^2)^3 = 6^{2*3} = 6^6$ , which is not equal to $6^5$ .
Evaluate option B: Evaluate option B. $\newline$ Option B is $2^5 \times 3^5$ . Since $6$ is the product of $2$ and $3$ , we can express $6^5$ as $(2\times3)^5$ . Using the distributive property of exponents over multiplication, $(2\times3)^5 = 2^5 \times 3^5$ . Therefore, option B is equivalent to $6^5$ .
Evaluate option C: Evaluate option C. $\newline$ Option C is $(6^6)/(6^1)$ . According to the rule of exponents for division, $a^m / a^n = a^{(m-n)}$ . So, $(6^6)/(6^1) = 6^{(6-1)} = 6^5$ , which is equal to the original expression.
Evaluate option D: Evaluate option D. $\newline$ Option D is $3^2 \times 2^3$ . This is not equivalent to $6^5$ because $3^2 \times 2^3 = 9 \times 8 = 72$ , which is not in the form of a single base raised to a single exponent and does not equal $6$ raised to any power.