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For any integer m and n, which of the following expression is equivalent to 25^(mn) ?
A. (5^(n))^(2m)
B. 5^(m)*5^(n)
C. 25^(m)*25^(n)
D. 25^(m)+25^(n)

For any integer $m$ and $n$ , which of the following expression is equivalent to $25^{mn}$ ? $\newline$ A. $(5^{n})^{2m}$ $\newline$ B. $5^{m}\cdot5^{n}$ $\newline$ C. $25^{m}\cdot25^{n}$ $\newline$ D. $25^{m}+25^{n}$

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Find derivatives of using multiple formulae

Full solution

Q. For any integer

m

and

n

, which of the following expression is equivalent to

25^{mn}

?

\newline

A.

(5^{n})^{2m}

\newline

B.

5^{m}\cdot5^{n}

\newline

C.

25^{m}\cdot25^{n}

\newline

D.

25^{m}+25^{n}

Express $25$ as Prime Factorization: We need to find an expression equivalent to $25^{mn}$ . Let's start by expressing $25$ in terms of its prime factorization. $\newline$ $25$ is $5$ squared, so $25 = 5^2$ . Therefore, $25^{mn}$ can be written as $(5^2)^{mn}$ .
Simplify Using Exponent Property: Using the property of exponents that $(a^b)^c = a^{b*c}$ , we can simplify $(5^2)^{mn}$ to $5^{2mn}$ .
Examine Answer Choices: Now let's examine each answer choice to see which one is equivalent to $5^{2mn}$ . $\newline$ A. $(5^n)^{2m}$ can be simplified using the property of exponents $(a^b)^c = a^{b*c}$ to $5^{n*2m} = 5^{2mn}$ , which looks like it could be equivalent to our expression. $\newline$ B. $5^m \times 5^n$ is using the property $a^m \times a^n = a^{m+n}$ , which would give us $5^{m+n}$ , not $5^{2mn}$ . $\newline$ C. $25^m \times 25^n$ is using the property $a^m \times a^n = a^{m+n}$ , which would give us $(5^n)^{2m}$ $0$ , not $(5^n)^{2m}$ $1$ . $\newline$ D. $(5^n)^{2m}$ $2$ is just a sum of two terms and does not use any property of exponents that would make it equivalent to $(5^n)^{2m}$ $1$ .
Analyze Answer Choices: From the above analysis, we can see that option A is the correct choice because $(5^n)^{(2m)}$ simplifies to $5^{2mn}$ , which is the same as our original expression $25^{mn}$ .

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