To divide the given expression [tex]\( \frac{20 m^2 n^3 - 10 m^2 n}{5 m n} \)[/tex], follow these steps:
1. Identify the numerator and denominator:
- The numerator is [tex]\( 20 m^2 n^3 - 10 m^2 n \)[/tex].
- The denominator is [tex]\( 5 m n \)[/tex].
2. Factor out common terms in the numerator:
- In the numerator [tex]\( 20 m^2 n^3 - 10 m^2 n \)[/tex], observe that both terms have common factors of [tex]\( 10 m^2 n \)[/tex].
- Factoring out [tex]\( 10 m^2 n \)[/tex] from the numerator, we get:
[tex]\[
20 m^2 n^3 - 10 m^2 n = 10 m^2 n (2 n^2 - 1)
\][/tex]
3. Rewrite the expression using the factored numerator:
- Substitute [tex]\( 10 m^2 n (2 n^2 - 1) \)[/tex] back into the numerator of the original fraction:
[tex]\[
\frac{10 m^2 n (2 n^2 - 1)}{5 m n}
\][/tex]
4. Simplify the fraction by cancelling common terms:
- Both the numerator and the denominator have [tex]\( 5 m n \)[/tex] as a common factor.
- Divide both the numerator and the denominator by [tex]\( 5 m n \)[/tex]:
[tex]\[
\frac{10 m^2 n (2 n^2 - 1)}{5 m n} = \frac{10 m^2 n (2 n^2 - 1)}{5 m n} = \frac{10 m^2}{5 m} \cdot \frac{n (2 n^2 - 1)}{n} = \frac{10}{5} \cdot m \cdot (2 n^2 - 1)
\][/tex]
5. Simplify the coefficients and terms:
- Simplify the coefficients: [tex]\( \frac{10}{5} = 2 \)[/tex]
- Simplify the [tex]\( m \)[/tex]-terms: [tex]\( m^2 / m = m \)[/tex]
So, the expression is:
[tex]\[
2 m (2 n^2 - 1)
\][/tex]
6. Write the final simplified expression:
- The final simplified expression is:
[tex]\[
4 m n^2 - 2 m
\][/tex]
Therefore, the result of dividing [tex]\( \frac{20 m^2 n^3 - 10 m^2 n}{5 m n} \)[/tex] is [tex]\( 4 m n^2 - 2 m \)[/tex].
