Engage with knowledgeable experts and get accurate answers on IDNLearn.com. Join our community to receive timely and reliable responses to your questions from knowledgeable professionals.

Select the correct answer.

Which expression is equivalent to the given expression?

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

A. [tex]\(4 m^{-1} n\)[/tex]

B. [tex]\(8 m^{-1} n\)[/tex]

C. [tex]\(4 m^9 n^3\)[/tex]

D. [tex]\(8 m^9 n^3\)[/tex]


Sagot :

To solve the given expression
[tex]\[ \frac{\left(4 m^2 n\right)^2}{2 m^5 n}, \][/tex]
we need to simplify it step by step.

### Step 1: Simplify the numerator

First, let's simplify the numerator [tex]\((4 m^2 n)^2\)[/tex].

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

Thus, the numerator simplifies to [tex]\( 16 m^4 n^2 \)[/tex].

### Step 2: Rewrite the expression with the simplified numerator

Now, substitute the simplified numerator back into the expression:
[tex]\[ \frac{16 m^4 n^2}{2 m^5 n} \][/tex]

### Step 3: Simplify the expression

Next, we need to divide the numerator by the denominator. Let's break it down:

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

Simplify each part separately:

- [tex]\(\frac{16}{2} = 8\)[/tex]
- [tex]\(\frac{m^4}{m^5} = m^{4-5} = m^{-1}\)[/tex]
- [tex]\(\frac{n^2}{n} = n^{2-1} = n\)[/tex]

### Step 4: Combine the simplified components

Putting it all together:

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

### Conclusion

The equivalent expression is [tex]\(8 m^{-1} n\)[/tex], which corresponds to option B.

So, the correct answer is:
[tex]\[ \boxed{B. \, 8 m^{-1} n} \][/tex]