Question

Perform the indicated operation:

Answer 1

The expression $\frac{m^{2}-16\cdot n^{2}}{\frac{3\cdot m + 12\cdot n}{m\cdot n} }$ is equal to $\frac{m^{2}\cdot n}{3} -\frac{4\cdot m\cdot n^{2}}{3}$.

How to simplify a composite algebraic expression

In this question we must use properties of real algebra to simplify a given expression as most as possible. We proceed to show each step with respective reason.

1. $\frac{m^{2}-16\cdot n^{2}}{\frac{3\cdot m + 12\cdot n}{m\cdot n} }$  Given
2. $\frac{m\cdot n \cdot (m^{2}-16\cdot n^{2})}{3\cdot m + 12\cdot n}$   $\frac{\frac{a}{b} }{\frac{c}{d} } = \frac{a\cdot d}{c\cdot b}$
3. $\frac{m\cdot n \cdot (m-4\cdot n)\cdot (m+4\cdot n)}{3\cdot (m+4\cdot n)}$ Distributive property/$a^{2}-b^{2} = (a +b)\cdot (a-b)$
4. $\frac{m\cdot n \cdot (m-4\cdot n)}{3}$ Existence of the multiplicative inverse/Modulative property
5. $\frac{m^{2}\cdot n}{3} -\frac{4\cdot m\cdot n^{2}}{3}$ Distributive property/$x^a\cdot x^{b} = x^{a+b}$/Result

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