91Ó°ÊÓ

When working with complex fractions, finding a common denominator is an essential first step. A common denominator is a shared multiple of the denominators of two or more fractions.
To simplify the given complex fraction \(\frac{s-r}{\frac{1}{r}-\frac{1}{s}}\), we first focus on the denominator: \(\frac{1}{r} - \frac{1}{s}\).
This means we need a common denominator for both \(r\) and \(s\).
A convenient choice here is the product of these two denominators, \(rs\). Rewrite each fraction with this shared denominator to combine them:
* \(\frac{1}{r}\) is re-written as \(\frac{s}{rs}\)
* \(\frac{1}{s}\) is re-written as \(\frac{r}{rs}\)
Combining these gives us the simplified form \(\frac{s-r}{rs}\). Understanding how to find and use common denominators is key to working with fractions efficiently.

Once you have your equations set up with a common denominator, the next step is understanding reciprocals.
A reciprocal of a fraction simply means flipping it upside down. For example, the reciprocal of \( \frac{a}{b} \) is \( \frac{b}{a} \).
This concept is useful especially when multiplying or simplifying complex fractions.
In our case, after combining the denominator fractions, we get \(\frac{s-r}{rs}\).
We now have \(\frac{s-r}{\frac{s-r}{rs}}\).
To simplify, we multiply by the reciprocal of the denominator fraction \(\frac{s-r}{rs}\), which is \(\frac{rs}{s-r}\). Now, the multiplication would look like:
\[\frac{s-r}{1} \times \frac{rs}{s-r} = rs\].
Whether dealing with simple or complex fractions, mastering reciprocals can simplify calculations greatly.

Simplifying fractions is the process of making them easier to work with.
For our exercise, this means reducing the complex fraction to a simpler form. After combining the fractions using a common denominator and understanding reciprocals, the final step is to simplify.
We start with \(\frac{s-r}{\frac{s-r}{rs}}\).
By applying the reciprocal, the complex fraction reduces to a simpler product of fractions as shown:
\[\frac{s-r}{1} \times \frac{rs}{s-r} = rs\].
By canceling out like terms \(s-r\) and multiplying the remaining terms, we simplify the entire expression down to just \(\rs\).
Simplifying fractions can significantly reduce the complexity of arithmetic and algebraic expressions, making them more manageable and easier to understand.