To divide the given expression [tex]\(\frac{15 m^4 n^3 - 35 m n^2}{5 m n}\)[/tex], follow these steps:
1. Rewrite the Numerator and Denominator:
The original problem is [tex]\(\frac{15 m^4 n^3 - 35 m n^2}{5 m n}\)[/tex].
2. Factor Out Common Terms in the Numerator:
We factor out the greatest common divisor (GCD) in the numerator. The GCD of the terms [tex]\(15 m^4 n^3\)[/tex] and [tex]\(35 m n^2\)[/tex] is [tex]\(5 m n^2\)[/tex]. Thus,
[tex]\[
15 m^4 n^3 - 35 m n^2 = 5 m n^2 (3 m^3 n - 7).
\][/tex]
3. Write the Expression:
Now the expression becomes:
[tex]\[
\frac{5 m n^2 (3 m^3 n - 7)}{5 m n}.
\][/tex]
4. Cancel the Common Factor:
You can cancel the common factor [tex]\(5 m n\)[/tex] in the numerator and the denominator:
[tex]\[
\frac{5 m n^2 (3 m^3 n - 7)}{5 m n} = \frac{n (m n) (3 m^3 n - 7)}{m n} = n (3 m^3 n - 7).
\][/tex]
5. Simplify the Expression:
After canceling the common terms, the simplified result is:
[tex]\[
n (3 m^3 n - 7).
\][/tex]
So the answer is:
[tex]\[
n (3 m^3 n - 7).
\][/tex]
Hence, [tex]\(\frac{15 m^4 n^3 - 35 m n^2}{5 m n}\)[/tex] simplifies correctly to [tex]\(\boxed{n (3 m^3 n - 7)}\)[/tex].
