To perform the operations on the given fractions, let's start by rewriting the expression with a common denominator. The expression is:
\[ \frac{4}{r^2 - 4r} + \frac{4}{r^2 + 8r} - \frac{1}{r^2 + 4r - 32} \]
First, we will factor the denominators:
- \( r^2 - 4r = r(r - 4) \)
- \( r^2 + 8r = r(r + 8) \)
- To factor \( r^2 + 4r - 32 \), we look for two numbers that multiply to \(-32\) and add to \(4\). These numbers are \(8\) and \(-4\), so: \[ r^2 + 4r - 32 = (r - 4)(r + 8) \]
Next, we rewrite our expression with the factored denominators:
\[ \frac{4}{r(r - 4)} + \frac{4}{r(r + 8)} - \frac{1}{(r - 4)(r + 8)} \]
Now, we need to find the least common denominator (LCD). The LCD will be:
\[ r(r - 4)(r + 8) \]
Now, we rewrite each fraction with this common denominator:
-
For the first term: \[ \frac{4}{r(r - 4)} = \frac{4(r + 8)}{r(r - 4)(r + 8)} \]
-
For the second term: \[ \frac{4}{r(r + 8)} = \frac{4(r - 4)}{r(r + 8)(r - 4)} \]
-
For the third term: \[ \frac{1}{(r - 4)(r + 8)} = \frac{r}{r(r - 4)(r + 8)} \]
Putting it all together, we have:
\[ \frac{4(r + 8) + 4(r - 4) - r}{r(r - 4)(r + 8)} \]
Now, simplify the numerator:
\[ 4(r + 8) + 4(r - 4) - r = 4r + 32 + 4r - 16 - r = 7r + 16 \]
Thus, we rewrite the expression:
\[ \frac{7r + 16}{r(r - 4)(r + 8)} \]
This is the simplified form of the original expression. Therefore, the final result is:
\[ \frac{7r + 16}{r(r - 4)(r + 8)} \]