91Ó°ÊÓ

When simplifying complex fractions, the first step is to simplify the numerator and the denominator separately. In the given exercise, we identify the numerator and the denominator of the complex fraction, \( \frac{\frac{1}{m+1}-1}{\frac{1}{m+1}+1} \).

For the numerator, \(\frac{1}{m+1} - 1\), we need a common denominator to combine the terms. By rewriting 1 as \(\frac{m+1}{m+1}\), we have:
\(\frac{1}{m+1} - \frac{m+1}{m+1} = \frac{1 - (m+1)}{m+1} = \frac{1 - m - 1}{m+1} = \frac{-m}{m+1}\).

Next, for the denominator, \(\frac{1}{m+1} + 1\), we do the same. By rewriting 1 as \(\frac{m+1}{m+1}\), we get:
\(\frac{1}{m+1} + \frac{m+1}{m+1} = \frac{1 + (m+1)}{m+1} = \frac{1 + m + 1}{m+1} = \frac{m+2}{m+1}\).

Now, both the numerator and the denominator are simplified fractions. This simplification is crucial because it prepares the fractions for the next steps: common denominator method and reciprocal multiplication.

Utilizing a common denominator helps to combine fractions more effectively. In our exercise, finding common denominators for both the numerator and the denominator fractions allows restructuring them as single fractions.

For the numerator, by writing 1 as \(\frac{m+1}{m+1}\), we were able to combine the fractions into \(\frac{-m}{m+1}\).

Similarly, for the denominator, by writing 1 as \(\frac{m+1}{m+1}\), we combined the fractions into \(\frac{m+2}{m+1}\).

This step is essential because operating with multiple fractions can be very confusing, especially in complex fractions. Combining them into single fractions significantly simplifies further calculations.

In summary, always look for ways to combine fractions by rewriting whole numbers or mixed fractions to have the same denominator. This process will make the subsequent mathematical operations much more manageable.

The final step in simplifying complex fractions involves dividing by the denominator, which is the same as multiplying by its reciprocal.

For our simplified complex fraction, \(\frac{\frac{-m}{m+1}}{\frac{m+2}{m+1}}\), dividing the numerator by the denominator can be changed to multiplying by the reciprocal of the denominator:
\(\frac{-m}{m+1} \times \frac{m+1}{m+2}\).
The \(m+1\) terms cancel out, simplifying the expression to:
\(\frac{-m}{m+2}\).

Reciprocal multiplication effectively transforms division into a multiplication problem, which is simpler to solve. This cancellation step removes common terms from the numerator and denominator, yielding the final reduced form.

In summary, the reciprocal multiplication method converts complex division into straightforward multiplication, streamlining the simplification process.