- The division of a rational expression by a term is equivalent to multiplying by the term's reciprocal: \(\frac{A}{B} \div C = \frac{A}{B} \cdot \frac{1}{C}\).
- Applying this rule, we rewrite the expression as \(\frac{m-4}{m+4} \cdot \frac{1}{m+2}\).
- Multiplying the numerators and denominators yields the simplified expression \(\frac{m-4}{(m+4)(m+2)}\).
- The final answer is \(\boxed{\frac{m-4}{(m+4)(m+2)}}\).
Explanation
- Understanding the Problem
The expression to simplify is \(\frac{m-4}{m+4} \div (m+2)\). This is a division problem involving a rational expression. In algebra, dividing by a value is equivalent to multiplying by its reciprocal. The reciprocal of a value \(x\) is \(\frac{1}{x}\), provided that \(x
eq 0\). Thus, our expression can be rewritten by replacing the division sign with a multiplication sign and taking the reciprocal of the term \((m+2)\).
- Simplifying the Expression
Applying the rule of division, we have:
\[\frac{m-4}{m+4} \div (m+2) = \frac{m-4}{m+4} \cdot \frac{1}{m+2}\]
Now, we perform the multiplication by multiplying the numerators together and the denominators together:
\[\frac{(m-4) \cdot 1}{(m+4) \cdot (m+2)} = \frac{m-4}{(m+4)(m+2)}\]
This results in a single rational expression that cannot be simplified further, as there are no common factors between the numerator and the denominator.
- Conclusion
Comparing our result \(\frac{m-4}{(m+4)(m+2)}\) to the provided choices:
- \(\frac{m-4}{(m+4)(m+2)}\)
- \(\frac{(m+4)(m+2)}{m-4}\)
- \(\frac{(m-4)(m+2)}{m+4}\)
- \(\frac{m+4}{(m-4)(m+2)}\)
Our calculated expression matches the first choice. Therefore, the equivalent expression is \(\boxed{\frac{m-4}{(m+4)(m+2)}}\).
Examples
Think of the expression \(\frac{m-4}{m+4} \div (m+2)\) as a way to distribute a set of items. Suppose you have a ratio of materials \(\frac{m-4}{m+4}\) representing the concentration of a certain substance in a solution, and you decide to split this entire concentration evenly across \(m+2\) different containers. By dividing the total concentration by the number of containers, you are finding the new concentration per container. This simple algebraic manipulation allows you to express complex ratios in a more manageable form, helping you calculate resources or distribute materials efficiently in fields like chemistry, cooking, or project management.