Answer :
To solve the problem of subtracting [tex]\(\frac{18r}{r^2-81} - \frac{9}{r-9}\)[/tex] and expressing the result as a single rational expression, let's follow these steps:
1. Factorize the Denominators:
- The first denominator is [tex]\(r^2 - 81\)[/tex]. Notice that this is a difference of squares:
[tex]\[ r^2 - 81 = (r - 9)(r + 9) \][/tex]
- The second denominator is already in a simple form: [tex]\(r - 9\)[/tex].
2. Rewrite Fractions with a Common Denominator:
- The common denominator for both fractions will be [tex]\((r - 9)(r + 9)\)[/tex]. We need to rewrite the second fraction so that it has this common denominator.
3. Rewrite the Second Fraction:
- For the fraction [tex]\(\frac{9}{r-9}\)[/tex], we can multiply the numerator and denominator by [tex]\((r + 9)\)[/tex] to match the common denominator:
[tex]\[ \frac{9}{r-9} = \frac{9(r + 9)}{(r - 9)(r + 9)} = \frac{9r + 81}{(r - 9)(r + 9)} \][/tex]
4. Combine the Fractions:
- Now that both fractions have the same denominator, we can combine them:
[tex]\[ \frac{18r}{(r - 9)(r + 9)} - \frac{9r + 81}{(r - 9)(r + 9)} \][/tex]
- Since the denominators are the same, we subtract the numerators:
[tex]\[ \frac{18r - (9r + 81)}{(r - 9)(r + 9)} \][/tex]
5. Simplify the Numerator:
- Distribute the negative sign in the numerator:
[tex]\[ 18r - 9r - 81 = 9r - 81 \][/tex]
- So our expression is:
[tex]\[ \frac{9r - 81}{(r - 9)(r + 9)} \][/tex]
6. Factor and Simplify:
- Notice that the numerator [tex]\(9r - 81\)[/tex] can be factored out:
[tex]\[ 9r - 81 = 9(r - 9) \][/tex]
- The expression becomes:
[tex]\[ \frac{9(r - 9)}{(r - 9)(r + 9)} \][/tex]
- Cancel out the common factor of [tex]\(r - 9\)[/tex]:
[tex]\[ \frac{9 \cancel{(r - 9)}}{\cancel{(r - 9)} (r + 9)} = \frac{9}{r + 9} \][/tex]
Therefore, the simplified single rational expression is:
[tex]\[ \boxed{\frac{9}{r + 9}} \][/tex]
1. Factorize the Denominators:
- The first denominator is [tex]\(r^2 - 81\)[/tex]. Notice that this is a difference of squares:
[tex]\[ r^2 - 81 = (r - 9)(r + 9) \][/tex]
- The second denominator is already in a simple form: [tex]\(r - 9\)[/tex].
2. Rewrite Fractions with a Common Denominator:
- The common denominator for both fractions will be [tex]\((r - 9)(r + 9)\)[/tex]. We need to rewrite the second fraction so that it has this common denominator.
3. Rewrite the Second Fraction:
- For the fraction [tex]\(\frac{9}{r-9}\)[/tex], we can multiply the numerator and denominator by [tex]\((r + 9)\)[/tex] to match the common denominator:
[tex]\[ \frac{9}{r-9} = \frac{9(r + 9)}{(r - 9)(r + 9)} = \frac{9r + 81}{(r - 9)(r + 9)} \][/tex]
4. Combine the Fractions:
- Now that both fractions have the same denominator, we can combine them:
[tex]\[ \frac{18r}{(r - 9)(r + 9)} - \frac{9r + 81}{(r - 9)(r + 9)} \][/tex]
- Since the denominators are the same, we subtract the numerators:
[tex]\[ \frac{18r - (9r + 81)}{(r - 9)(r + 9)} \][/tex]
5. Simplify the Numerator:
- Distribute the negative sign in the numerator:
[tex]\[ 18r - 9r - 81 = 9r - 81 \][/tex]
- So our expression is:
[tex]\[ \frac{9r - 81}{(r - 9)(r + 9)} \][/tex]
6. Factor and Simplify:
- Notice that the numerator [tex]\(9r - 81\)[/tex] can be factored out:
[tex]\[ 9r - 81 = 9(r - 9) \][/tex]
- The expression becomes:
[tex]\[ \frac{9(r - 9)}{(r - 9)(r + 9)} \][/tex]
- Cancel out the common factor of [tex]\(r - 9\)[/tex]:
[tex]\[ \frac{9 \cancel{(r - 9)}}{\cancel{(r - 9)} (r + 9)} = \frac{9}{r + 9} \][/tex]
Therefore, the simplified single rational expression is:
[tex]\[ \boxed{\frac{9}{r + 9}} \][/tex]
