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If
2^(m)=x and
7^(n)=y, then which of the following is equivalent to
784^(mn) ?
Choose 1 answer:
(A)
x^(2m)*y^(4n)
(B)
x^(2n)*y^(4m)
(C)
x^(4m)*y^(2n)
(D)
x^(4n)*y^(2m)

If $2^{m}=x$ and $7^{n}=y$ , then which of the following is equivalent to $784^{m n}$ ? $\newline$ Choose $1$ answer: $\newline$ (A) $x^{2 m} \cdot y^{4 n}$ $\newline$ (B) $x^{2 n} \cdot y^{4 m}$ $\newline$ (C) $x^{4 m} \cdot y^{2 n}$ $\newline$ (D) $x^{4 n} \cdot y^{2 m}$

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Compare linear and exponential growth

Full solution

Q. If

2^{m}=x

and

7^{n}=y

, then which of the following is equivalent to

784^{m n}

?

\newline

Choose

1

answer:

\newline

(A)

x^{2 m} \cdot y^{4 n}

\newline

(B)

x^{2 n} \cdot y^{4 m}

\newline

(C)

x^{4 m} \cdot y^{2 n}

\newline

(D)

x^{4 n} \cdot y^{2 m}

Expressing $784$ as Prime Factors: First, let's express $784$ as a product of its prime factors. $\newline$ $784 = 2^4 \times 7^2$ $\newline$ This is because $784$ is divisible by $2$ four times ( $2 \times 2 \times 2 \times 2 = 16$ ) and by $7$ twice ( $7 \times 7 = 49$ ), and $16 \times 49 = 784$ .
Raising Prime Factorization to Power: Now, let's raise this prime factorization to the power of $mn$ : $\newline$ $(2^4 \times 7^2)^{mn} = (2^{4mn}) \times (7^{2mn})$ $\newline$ We distribute the exponent $mn$ to both $2^4$ and $7^2$ .
Substituting Variables: Next, we substitute $x$ for $2^m$ and $y$ for $7^n$ according to the given information: $\newline$ $2^{m} = x$ and $7^{n} = y$ $\newline$ So, $2^{4mn}$ becomes $x^{4n}$ and $7^{2mn}$ becomes $y^{2m}$ .
Writing the Expression with Substituted Variables: Now, we write the expression with the substituted variables: $\newline$ $(2^{4mn}) \times (7^{2mn}) = x^{4n} \times y^{2m}$ $\newline$ This matches one of the answer choices.
Comparing with Answer Choices: We compare our expression with the answer choices: $\newline$ (A) $x^{2m} \cdot y^{4n}$ $\newline$ (B) $x^{2n} \cdot y^{4m}$ $\newline$ (C) $x^{4m} \cdot y^{2n}$ $\newline$ (D) $x^{4n} \cdot y^{2m}$ $\newline$ Our expression is $x^{4n} \cdot y^{2m}$ , which matches answer choice (D).

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