13) $\dfrac{\left(\dfrac{1}{a - b} - \dfrac{3}{a + b}\
ight)}{\left(\dfrac{2}{b - a} + \dfrac{4}{b + a}\
ight)}$

Question

Accepted Answer

# Explanation:
## Step1: Simplify numerator
First, simplify the numerator $\frac{1}{a - b}-\frac{3}{a + b}$. Find a common denominator $(a - b)(a + b)$:
$$
\begin{align*}
\frac{1}{a - b}-\frac{3}{a + b}&=\frac{(a + b)-3(a - b)}{(a - b)(a + b)}\
&=\frac{a + b-3a + 3b}{(a - b)(a + b)}\
&=\frac{-2a + 4b}{(a - b)(a + b)}
\end{align*}
$$
## Step2: Simplify denominator
Simplify the denominator $\frac{2}{b - a}+\frac{4}{b + a}$. Note that $b - a=-(a - b)$, so:
$$
\begin{align*}
\frac{2}{b - a}+\frac{4}{b + a}&=\frac{2}{-(a - b)}+\frac{4}{a + b}\
&=-\frac{2}{a - b}+\frac{4}{a + b}\
&=\frac{-2(a + b)+4(a - b)}{(a - b)(a + b)}\
&=\frac{-2a-2b + 4a-4b}{(a - b)(a + b)}\
&=\frac{2a-6b}{(a - b)(a + b)}
\end{align*}
$$
## Step3: Divide numerator by denominator
Now, divide the simplified numerator by the simplified denominator:
$$
\begin{align*}
\frac{\frac{-2a + 4b}{(a - b)(a + b)}}{\frac{2a-6b}{(a - b)(a + b)}}&=\frac{-2a + 4b}{(a - b)(a + b)}\times\frac{(a - b)(a + b)}{2a-6b}\
&=\frac{-2(a - 2b)}{2(a - 3b)}\
&=\frac{-(a - 2b)}{a - 3b}\
&=\frac{-a + 2b}{a - 3b}\
&=\frac{2b - a}{a - 3b}
\end{align*}
$$
# Answer:
$\frac{2b - a}{a - 3b}$

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QUESTION IMAGE

Question

Explanation:

Step1: Simplify numerator

Step2: Simplify denominator

Step3: Divide numerator by denominator

Answer: