To simplify the given expression [tex]\(\frac{2m^5 n^2 - 6m^2 n^3}{3mn}\)[/tex], follow these steps:
1. Factor the Numerator:
The numerator of the expression is [tex]\(2m^5 n^2 - 6m^2 n^3\)[/tex]. Notice that there is a common factor in both terms:
- The greatest common factor (GCF) is [tex]\(2m^2n^2\)[/tex].
- When you factor this out from each term in the numerator, you get:
[tex]\[
2m^5 n^2 - 6m^2 n^3 = 2m^2n^2 (m^3) - 2m^2n^2 (3n) = 2m^2n^2 (m^3 - 3n)
\][/tex]
2. Rewrite the Expression with the Common Factor:
By factoring out the common factor, the expression becomes:
[tex]\[
\frac{2m^2 n^2 (m^3 - 3n)}{3mn}
\][/tex]
3. Simplify the Denominator:
Notice that the denominator is simply [tex]\(3mn\)[/tex], as given.
4. Cancel Out Common Factors:
In the fraction [tex]\(\frac{2m^2 n^2 (m^3 - 3n)}{3mn}\)[/tex], you can cancel out the common factors in the numerator and the denominator:
- [tex]\(m\)[/tex] in the numerator's factor [tex]\(m^2 n^2\)[/tex] and the denominator [tex]\(3mn\)[/tex]:
[tex]\[
\frac{2m^2 n^2 (m^3 - 3n)}{3mn} = \frac{2m n^2 (m^3 - 3n)}{3n}
\][/tex]
- [tex]\(n\)[/tex] in the numerator's factor [tex]\(n^2\)[/tex] and the denominator [tex]\(3n\)[/tex]:
[tex]\[
\frac{2m n^2 (m^3 - 3n)}{3n} = \frac{2m n (m^3 - 3n)}{3}
\][/tex]
5. Write the Final Simplified Expression:
The simplified form of the original expression is:
[tex]\[
\frac{2m n (m^3 - 3n)}{3}
\][/tex]
So, the simplified expression is:
[tex]\[
\boxed{\frac{2m n (m^3 - 3n)}{3}}
\][/tex]
