Math Calculators, Lessons and Formulas

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Answer

$$\frac{\frac{r}{s}-\frac{s}{r}}{\frac{1}{s}+\frac{1}{r}}=\frac{r^2-s^2}{r+s}$$

Explanation

Tap the blue circles to see an explanation.

$$ \begin{aligned}\frac{\frac{r}{s}-\frac{s}{r}}{\frac{1}{s}+\frac{1}{r}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{\frac{r^2-s^2}{rs}}{\frac{r+s}{rs}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{r^2-s^2}{r+s}\end{aligned} $$
①

Subtract $ \dfrac{s}{r} $ from $ \dfrac{r}{s} $ to get $ \dfrac{ \color{purple}{ r^2-s^2 } }{ rs }$.

To subtract raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $ \color{blue}{ r }$ and the second by $\color{blue}{ s }$.

$$ \begin{aligned} \frac{r}{s} - \frac{s}{r} & = \frac{ r \cdot \color{blue}{ r }}{ s \cdot \color{blue}{ r }} - \frac{ s \cdot \color{blue}{ s }}{ r \cdot \color{blue}{ s }} = \\[1ex] &=\frac{ \color{purple}{ r^2 } }{ rs } - \frac{ \color{purple}{ s^2 } }{ rs }=\frac{ \color{purple}{ r^2-s^2 } }{ rs } \end{aligned} $$
②

Add $ \dfrac{1}{s} $ and $ \dfrac{1}{r} $ to get $ \dfrac{ \color{purple}{ r+s } }{ rs }$.

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}{ r }$ and the second by $\color{blue}{ s }$.

$$ \begin{aligned} \frac{1}{s} + \frac{1}{r} & = \frac{ 1 \cdot \color{blue}{ r }}{ s \cdot \color{blue}{ r }} + \frac{ 1 \cdot \color{blue}{ s }}{ r \cdot \color{blue}{ s }} = \\[1ex] &=\frac{ \color{purple}{ r } }{ rs } + \frac{ \color{purple}{ s } }{ rs }=\frac{ \color{purple}{ r+s } }{ rs } \end{aligned} $$
③

Divide $ \dfrac{r^2-s^2}{rs} $ by $ \dfrac{r+s}{rs} $ to get $ \dfrac{ r^2-s^2 }{ r+s } $.

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

Step 2: Cancel $ \color{red}{ rs } $ in first and second fraction.

Step 3: Multiply numerators and denominators.

Step 4: Simplify numerator and denominator.

$$ \begin{aligned} \frac{ \frac{r^2-s^2}{rs} }{ \frac{\color{blue}{r+s}}{\color{blue}{rs}} } & \xlongequal{\text{Step 1}} \frac{r^2-s^2}{rs} \cdot \frac{\color{blue}{rs}}{\color{blue}{r+s}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{r^2-s^2}{\color{red}{1}} \cdot \frac{\color{red}{1}}{r+s} \xlongequal{\text{Step 3}} \frac{ \left( r^2-s^2 \right) \cdot 1 }{ 1 \cdot \left( r+s \right) } = \\[1ex] & \xlongequal{\text{Step 4}} \frac{ r^2-s^2 }{ r+s } \end{aligned} $$
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