Absolutely, let's simplify the given expression step-by-step:
We start with the expression:
[tex]\[
\frac{10 m^5 n^2 - 30 m n}{5 m n}
\][/tex]
Step 1: Factor the numerator and the denominator separately.
The numerator is:
[tex]\[
10 m^5 n^2 - 30 m n
\][/tex]
We can factor out the greatest common factor (GCF). The GCF in the numerator is [tex]\(10 m n\)[/tex], so we have:
[tex]\[
10 m^5 n^2 - 30 m n = 10 m n (m^4 n - 3)
\][/tex]
The denominator is:
[tex]\[
5 m n
\][/tex]
Step 2: Write the expression with the factored form of the numerator.
[tex]\[
\frac{10 m n (m^4 n - 3)}{5 m n}
\][/tex]
Step 3: Cancel out the common factor in the numerator and the denominator.
[tex]\(5 m n\)[/tex] is a common factor in both the numerator and the denominator:
[tex]\[
\frac{10 m n (m^4 n - 3)}{5 m n} = \frac{10}{5} \cdot \frac{m n}{m n} \cdot (m^4 n - 3)
\][/tex]
Step 4: Simplify the fraction.
[tex]\[
\frac{10}{5} = 2
\][/tex]
[tex]\[
\frac{m n}{m n} = 1
\][/tex]
Thus, we are left with:
[tex]\[
2 \cdot (m^4 n - 3)
\][/tex]
Step 5: Distribute the 2.
[tex]\[
2 \cdot (m^4 n - 3) = 2 m^4 n - 6
\][/tex]
So, the simplified form of the given expression [tex]\(\frac{10 m^5 n^2 - 30 m n}{5 m n}\)[/tex] is:
[tex]\[
2 m^4 n - 6
\][/tex]
