Which expression is equivalent to $4^6 \times 9^6$ ? $\newline$ Choices: $\newline$ (A) $36^{12}$ $\newline$ (B) $36^{36}$ $\newline$ (C) $36^6$ $\newline$ (D) $1/36^6$

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Grade 8

Identify equivalent expressions involving exponents I

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Q. Which expression is equivalent to

4^6 \times 9^6

\newline

Choices:

\newline

(A)

36^{12}

\newline

(B)

36^{36}

\newline

(C)

36^6

\newline

(D)

1/36^6

Identify Bases and Exponents: Identify the base and the exponents in the given expression. $\newline$ The given expression is $4^6 \times 9^6$ . Here, $4$ and $9$ are the bases, and both are raised to the exponent $6$ .
Recognize Perfect Squares: Recognize that $4$ and $9$ are perfect squares. $4$ is the square of $2$ (since $2^2 = 4$ ), and $9$ is the square of $3$ (since $3^2 = 9$ ). This can be rewritten as $(2^2)^6 \times (3^2)^6$ .
Apply Power of Power Rule: Apply the power of a power rule. $\newline$ According to the power of a power rule, $(a^b)^c = a^{(b*c)}$ . Therefore, $(2^2)^6$ becomes $2^{(2*6)}$ and $(3^2)^6$ becomes $3^{(2*6)}$ .
Calculate New Exponents: Calculate the new exponents. $2^{(2*6)} = 2^{12}$ and $3^{(2*6)} = 3^{12}$ . So the expression now is $2^{12} * 3^{12}$ .
Combine Bases with Multiplication: Combine the bases using multiplication. $\newline$ Since both bases are raised to the same exponent, we can multiply the bases together and keep the common exponent. This gives us $(2\times3)^{12}$ .
Calculate Base Multiplication: Calculate the base multiplication. $2 \times 3$ equals $6$ . So the expression now is $6^{12}$ .
Recognize Product of $4$ and $9$ : Recognize that $6$ is the product of $4$ and $9$ . $\newline$ Since $4\times9$ equals $36$ , and we have $6^{12}$ , we can rewrite this as $36^6$ .
Choose Equivalent Expression: Choose the equivalent expression for $4^6 \times 9^6$ . The equivalent expression is $36^6$ , which corresponds to choice (C).