Question

subtract the following fractions. simplify your answer as much as possible.
\\(\frac{8r}{r^2 - 16} - \frac{4}{r - 4}\\)

Explanation:

Step1: Factor the denominator

First, factor \( r^2 - 16 \) using the difference of squares formula \( a^2 - b^2=(a + b)(a - b) \). So \( r^2 - 16=(r + 4)(r - 4) \). The expression becomes \( \frac{8r}{(r + 4)(r - 4)}-\frac{4}{r - 4} \).

Step2: Find a common denominator

The common denominator of \( (r + 4)(r - 4) \) and \( r - 4 \) is \( (r + 4)(r - 4) \). Rewrite the second fraction with the common denominator: \( \frac{4}{r - 4}=\frac{4(r + 4)}{(r + 4)(r - 4)} \). Now the expression is \( \frac{8r}{(r + 4)(r - 4)}-\frac{4(r + 4)}{(r + 4)(r - 4)} \).

Step3: Subtract the numerators

Subtract the numerators over the common denominator: \( \frac{8r-4(r + 4)}{(r + 4)(r - 4)} \). Expand the numerator: \( 8r-4r - 16 = 4r-16 \). So the expression becomes \( \frac{4r - 16}{(r + 4)(r - 4)} \).

Step4: Simplify the numerator

Factor out 4 from the numerator: \( \frac{4(r - 4)}{(r + 4)(r - 4)} \).

Step5: Cancel common factors

Cancel out the common factor \( r - 4 \) (assuming \( r
eq4 \)): \( \frac{4}{r + 4} \).

Answer:

\(\frac{4}{r + 4}\)