Which expression is equivalent to $9^4 \times 4^4$ ? $\newline$ Choices: $\newline$ (A) $36^8$ $\newline$ (B) $36^4$ $\newline$ (C) $1/36^4$ $\newline$ (D) $36^{16}$

View step-by-step help

Home

Math Problems

Grade 8

Identify equivalent expressions involving exponents I

Full solution

Q. Which expression is equivalent to

9^4 \times 4^4

\newline

Choices:

\newline

(A)

36^8

\newline

(B)

36^4

\newline

(C)

1/36^4

\newline

(D)

36^{16}

Identify Bases and Exponents: Identify the base numbers and their exponents in the given expression. $\newline$ The given expression is $9^4 \times 4^4$ . Here, $9$ and $4$ are the bases, and both are raised to the power of $4$ .
Factor Bases to Common Base: Factor the bases to express them as powers of a common base if possible. $\newline$ The number $9$ can be written as $3^2$ , and the number $4$ can be written as $2^2$ . Therefore, we can rewrite the expression as $(3^2)^4 \times (2^2)^4$ .
Apply Power of Power Rule: Apply the power of a power rule, which states that a^b)^c = a^{b*c}\. Using this rule, we get \(\(3^ $2$ )^ $4$ * ( $2$ ^ $2$ )^ $4$ = $3$ ^{ $2$ * $4$ } * $2$ ^{ $2$ * $4$ } = $3$ ^ $8$ * $2$ ^ $8$ \
Multiply Bases with Exponents: Multiply the expressions with the same exponents by combining the bases. $\newline$ Since both $3^8$ and $2^8$ have the same exponent, we can multiply the bases first and then apply the exponent. This gives us $(3 \times 2)^8 = 6^8$ .
Factor $6$ and Simplify: Recognize that $6^8$ is not one of the answer choices, but $6$ can be factored further. $\newline$ The number $6$ can be written as $2 \times 3$ , and since we already have $2^8$ and $3^8$ from the previous step, we can combine them to get $(2 \times 3)^8 = 6^8$ .
Express in Terms of Base^$4$: Realize that $6^8$ is not in the simplest form for comparison with the answer choices. $\newline$ We need to express $6^8$ in terms of a base raised to the power of $4$ to match the answer choices. Since $6$ is $2\times3$ , we can write $6^8$ as $(2^4 \times 3^4)^2$ .
Apply Distributive Property: Apply the distributive property of exponents over multiplication. $\newline$ We have $(2^4 \times 3^4)^2$ , which can be written as $(2^4)^2 \times (3^4)^2$ . This simplifies to $2^8 \times 3^8$ , which is the same as $6^8$ from the previous steps.
Recognize Equivalent Expression: Recognize that $2^8 \times 3^8$ is equivalent to $(2\times3)^8$ , which is $6^8$ . We have already established that $6^8$ is the simplified form of the given expression. Now we need to express it in terms of a base raised to the power of $4$ to match the answer choices.
Express in Terms of Base^$4$: Express $6^8$ in terms of a base raised to the power of $4$ . $\newline$ Since $6^8 = (6^2)^4$ , and $6^2$ is $36$ , we can write $6^8$ as $36^4$ .
Match to Answer Choices: Match the simplified expression to the answer choices. $\newline$ The expression $36^4$ corresponds to choice (B) $36^4$ .