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Answer

$$\frac{\frac{5}{p+7}+\frac{9}{p-7}}{\frac{1}{p^2-49}+\frac{7}{p+7}}=\frac{14p+28}{7p-48}$$

Explanation

Tap the blue circles to see an explanation.

$$ \begin{aligned}\frac{\frac{5}{p+7}+\frac{9}{p-7}}{\frac{1}{p^2-49}+\frac{7}{p+7}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{\frac{14p+28}{p^2-49}}{\frac{7p-48}{p^2-49}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{14p+28}{7p-48}\end{aligned} $$
①

Add $ \dfrac{5}{p+7} $ and $ \dfrac{9}{p-7} $ to get $ \dfrac{ \color{purple}{ 14p+28 } }{ p^2-49 }$.

To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $ \color{blue}{ p-7 }$ and the second by $\color{blue}{ p+7 }$.

$$ \begin{aligned} \frac{5}{p+7} + \frac{9}{p-7} & = \frac{ 5 \cdot \color{blue}{ \left( p-7 \right) }}{ \left( p+7 \right) \cdot \color{blue}{ \left( p-7 \right) }} + \frac{ 9 \cdot \color{blue}{ \left( p+7 \right) }}{ \left( p-7 \right) \cdot \color{blue}{ \left( p+7 \right) }} = \\[1ex] &=\frac{ \color{purple}{ 5p-35 } }{ p^2 -\cancel{7p}+ \cancel{7p}-49 } + \frac{ \color{purple}{ 9p+63 } }{ p^2 -\cancel{7p}+ \cancel{7p}-49 } = \\[1ex] &=\frac{ \color{purple}{ 14p+28 } }{ p^2-49 } \end{aligned} $$
②

Add $ \dfrac{1}{p^2-49} $ and $ \dfrac{7}{p+7} $ to get $ \dfrac{ \color{purple}{ 7p-48 } }{ p^2-49 }$.

To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{p-7}$.

$$ \begin{aligned} \frac{1}{p^2-49} + \frac{7}{p+7} & = \frac{ 1 }{ p^2-49 } + \frac{ 7 \cdot \color{blue}{ \left( p-7 \right) }}{ \left( p+7 \right) \cdot \color{blue}{ \left( p-7 \right) }} = \\[1ex] &=\frac{ \color{purple}{ 1 } }{ p^2-49 } + \frac{ \color{purple}{ 7p-49 } }{ p^2-49 }=\frac{ \color{purple}{ 7p-48 } }{ p^2-49 } \end{aligned} $$
③

Divide $ \dfrac{14p+28}{p^2-49} $ by $ \dfrac{7p-48}{p^2-49} $ to get $ \dfrac{ 14p+28 }{ 7p-48 } $.

Step 1: To divide rational expressions, multiply the first fraction by the reciprocal of the second fraction.

Step 2: Cancel $ \color{red}{ p^2-49 } $ in first and second fraction.

Step 3: Multiply numerators and denominators.

Step 4: Simplify numerator and denominator.

$$ \begin{aligned} \frac{ \frac{14p+28}{p^2-49} }{ \frac{\color{blue}{7p-48}}{\color{blue}{p^2-49}} } & \xlongequal{\text{Step 1}} \frac{14p+28}{p^2-49} \cdot \frac{\color{blue}{p^2-49}}{\color{blue}{7p-48}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{14p+28}{\color{red}{1}} \cdot \frac{\color{red}{1}}{7p-48} \xlongequal{\text{Step 3}} \frac{ \left( 14p+28 \right) \cdot 1 }{ 1 \cdot \left( 7p-48 \right) } = \\[1ex] & \xlongequal{\text{Step 4}} \frac{ 14p+28 }{ 7p-48 } \end{aligned} $$
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