91Ó°ÊÓ

To add or subtract fractions, you need a common denominator. The least common denominator (LCD) is the smallest expression that can be used as a common denominator for all the fractions involved.

In this exercise, we have the fractions \( \frac{6}{m+4} \) and \( \frac{9}{m-8} \). Their denominators are \( m+4 \) and \( m-8 \), respectively.

To find the LCD, multiply these denominators together: \[ (m+4)(m-8) \].
This product is the smallest expression that can be used as a common denominator for both fractions.

Using the LCD makes the process of combining fractions straightforward. It allows you to rewrite each fraction so that both have the same denominator, which is crucial for addition or subtraction.

Once you have the least common denominator (LCD), the next step is to rewrite each fraction with this common denominator.

For \( \frac{6}{m+4} \), we rewrite it as follows:

\[ \frac{6}{m+4} = \frac{6(m-8)}{(m+4)(m-8)} \]

Similarly, for \( \frac{9}{m-8} \), we rewrite it as:

\[ \frac{9}{m-8} = \frac{9(m+4)}{(m+4)(m-8)} \]

Now, both fractions have the same denominator \[ (m+4)(m-8) \].
You can now combine them into a single fraction:

\[ \frac{6(m-8)}{(m+4)(m-8)} + \frac{9(m+4)}{(m+4)(m-8)} = \frac{6(m-8) + 9(m+4)}{(m+4)(m-8)} \]

Combining fractions this way ensures that all parts of the expression are considered and incorporated.

After combining the fractions, the next step is simplifying the algebraic expression in the numerator.

In our example, the combined fraction is:

\[ \frac{6(m-8) + 9(m+4)}{(m+4)(m-8)} \].

Simplify the numerator by distributing and combining like terms:

\[ 6(m-8) + 9(m+4) = 6m - 48 + 9m + 36 \].

Combine the \( m \) terms and the constants:

\[ 6m + 9m - 48 + 36 = 15m - 12 \].

This gives us a simplified numerator: \[ 15m - 12 \].

Doing this step ensures that the fraction is in its simplest possible form before writing the final answer.

Rational expressions are fractions where the numerator and the denominator are both polynomials.

Simplifying rational expressions usually involves factoring, finding the least common denominator, and combining fractions.

In this exercise, the given expression is:

\[ \frac{6}{m+4} + \frac{9}{m-8} \].

By following the steps outlined—finding the LCD, rewriting fractions with the LCD, combining the fractions, and simplifying the numerator—we simplify the expression.

The final simplified form is:

\[ \frac{15m-12}{(m+4)(m-8)} \].

Understanding how to handle rational expressions is essential, as they appear frequently in algebra.
Simplifying them correctly will help solve more complex problems later on.