To simplify the given expression, [tex]\(\frac{-9 m n^2}{3 m^4 n}\)[/tex], we will follow a systematic process. Let’s break it down step-by-step:
1. Separate the Constants and Variables:
The numerator is [tex]\(-9 m n^2\)[/tex], and the denominator is [tex]\(3 m^4 n\)[/tex].
First, divide the constants:
[tex]\[
\frac{-9}{3} = -3
\][/tex]
Now, the expression becomes:
[tex]\[
-3 \cdot \frac{m n^2}{m^4 n}
\][/tex]
2. Simplify the [tex]\(m\)[/tex] Terms:
In the fraction [tex]\(\frac{m n^2}{m^4 n}\)[/tex], the [tex]\(m\)[/tex] terms are [tex]\(m\)[/tex] in the numerator and [tex]\(m^4\)[/tex] in the denominator. Simplify by subtracting the powers (using [tex]\(\frac{m^a}{m^b} = m^{a-b}\)[/tex]):
[tex]\[
\frac{m}{m^4} = m^{1-4} = m^{-3}
\][/tex]
3. Simplify the [tex]\(n\)[/tex] Terms:
Similarly, for the [tex]\(n\)[/tex] terms, we have [tex]\(n^2\)[/tex] in the numerator and [tex]\(n\)[/tex] in the denominator:
[tex]\[
\frac{n^2}{n} = n^{2-1} = n
\][/tex]
4. Combine the Simplified Parts:
Now, combine the simplified [tex]\(m\)[/tex] and [tex]\(n\)[/tex] terms with the constant:
[tex]\[
-3 \cdot m^{-3} \cdot n
\][/tex]
So, the simplified form of the expression [tex]\(\frac{-9 m n^2}{3 m^4 n}\)[/tex] is:
[tex]\[
-3 \cdot n \cdot m^{-3} \quad \text{or} \quad \frac{-3n}{m^3}
\][/tex]
Among the given options, this matches with:
[tex]\[
-3 \frac{n}{m^3}
\][/tex]
Thus, the correct simplified form is [tex]\(\frac{n}{m^3}\)[/tex]. However, it seems there’s a mismatch with the final format you may want to align it with exactly these. Thus, the choice matching the simplification correctly is [tex]\(\boxed{-3 \frac{n}{m^3}}\)[/tex].
