Which expression is equivalent to the given expression?

[tex]\[
\frac{(4m^2 n)^2}{2m^5 n}
\][/tex]

A. [tex]\(8m^{-1} n\)[/tex]
B. [tex]\(4m^{-1} n\)[/tex]
C. [tex]\(4m^9 n^3\)[/tex]
D. [tex]\(8m^9 n^3\)[/tex]



Answer :

Let's solve the given expression step by step to find the equivalent expression. The expression provided is:

[tex]\[ \frac{(4 m^2 n)^2}{2 m^5 n} \][/tex]

First, we need to simplify the numerator [tex]\((4 m^2 n)^2\)[/tex]:

[tex]\[ (4 m^2 n)^2 = (4)^2 \cdot (m^2)^2 \cdot (n)^2 = 16 m^4 n^2 \][/tex]

So, the expression now looks like:

[tex]\[ \frac{16 m^4 n^2}{2 m^5 n} \][/tex]

Next, we simplify this fraction by dividing both the numerator and the denominator by common factors.

Start by simplifying the constants:
[tex]\[ \frac{16}{2} = 8 \][/tex]

Now we have:
[tex]\[ \frac{8 m^4 n^2}{m^5 n} \][/tex]

To further simplify this, we divide each variable (and their exponents) in the numerator by the corresponding variable (and their exponents) in the denominator.

For [tex]\(m\)[/tex]:
[tex]\[ \frac{m^4}{m^5} = m^{4-5} = m^{-1} \][/tex]

For [tex]\(n\)[/tex]:
[tex]\[ \frac{n^2}{n} = n^{2-1} = n \][/tex]

Putting it all together, we have:

[tex]\[ 8 m^{-1} n \][/tex]

Hence, the correct expression equivalent to the given expression is:

[tex]\[ 8 m^{-1} n \][/tex]

The correct choice is:
A. [tex]\(8 m^{-1} n\)[/tex]