Let's simplify the given algebraic expression step-by-step:
Given expression:
[tex]\[ \frac{36mn \cdot (25m^2 - 49n^2)}{4mn \cdot (5m + 7n)} \][/tex]
First, factorize the numerator and the denominator where possible.
1. Recognize that [tex]\(25m^2 - 49n^2\)[/tex] is a difference of squares. Recall that:
[tex]\[ a^2 - b^2 = (a - b)(a + b) \][/tex]
In this case, [tex]\(a = 5m\)[/tex] and [tex]\(b = 7n\)[/tex], so:
[tex]\[ 25m^2 - 49n^2 = (5m)^2 - (7n)^2 = (5m + 7n)(5m - 7n) \][/tex]
Thus, the expression becomes:
[tex]\[ \frac{36mn \cdot (5m + 7n)(5m - 7n)}{4mn \cdot (5m + 7n)} \][/tex]
2. Cancel out the common factors in the numerator and the denominator.
- Notice that [tex]\(mn\)[/tex] appears in both the numerator and the denominator and can be cancelled out.
- Similarly, [tex]\((5m + 7n)\)[/tex] appears in both the numerator and the denominator and can be cancelled out as well.
After cancelling the common factors, we get:
[tex]\[ \frac{36 \cdot (5m - 7n)}{4} \][/tex]
3. Simplify the remaining numerical coefficient:
[tex]\[ \frac{36}{4} = 9 \][/tex]
So the expression reduces to:
[tex]\[ 9 \cdot (5m - 7n) \][/tex]
4. Distribute the 9:
[tex]\[ 9(5m - 7n) = 45m - 63n \][/tex]
Thus, the simplified form of the given expression is:
[tex]\[ \boxed{45m - 63n} \][/tex]
