Solve r/s+s/r/frac{r^2{s^2}-s^2/r^2}

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

Evaluate

$$\frac{rs}{r^{2}-s^{2}}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of s and r is rs. Multiply \frac{r}{s} times \frac{r}{r}. Multiply \frac{s}{r} times \frac{s}{s}.

$$\frac{\frac{rr}{rs}+\frac{ss}{rs}}{\frac{r^{2}}{s^{2}}-\frac{s^{2}}{r^{2}}}$$

Since \frac{rr}{rs} and \frac{ss}{rs} have the same denominator, add them by adding their numerators.

$$\frac{\frac{rr+ss}{rs}}{\frac{r^{2}}{s^{2}}-\frac{s^{2}}{r^{2}}}$$

Do the multiplications in rr+ss.

$$\frac{\frac{r^{2}+s^{2}}{rs}}{\frac{r^{2}}{s^{2}}-\frac{s^{2}}{r^{2}}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of s^{2} and r^{2} is r^{2}s^{2}. Multiply \frac{r^{2}}{s^{2}} times \frac{r^{2}}{r^{2}}. Multiply \frac{s^{2}}{r^{2}} times \frac{s^{2}}{s^{2}}.

$$\frac{\frac{r^{2}+s^{2}}{rs}}{\frac{r^{2}r^{2}}{r^{2}s^{2}}-\frac{s^{2}s^{2}}{r^{2}s^{2}}}$$

Since \frac{r^{2}r^{2}}{r^{2}s^{2}} and \frac{s^{2}s^{2}}{r^{2}s^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{r^{2}+s^{2}}{rs}}{\frac{r^{2}r^{2}-s^{2}s^{2}}{r^{2}s^{2}}}$$

Do the multiplications in r^{2}r^{2}-s^{2}s^{2}.

$$\frac{\frac{r^{2}+s^{2}}{rs}}{\frac{r^{4}-s^{4}}{r^{2}s^{2}}}$$

Divide \frac{r^{2}+s^{2}}{rs} by \frac{r^{4}-s^{4}}{r^{2}s^{2}} by multiplying \frac{r^{2}+s^{2}}{rs} by the reciprocal of \frac{r^{4}-s^{4}}{r^{2}s^{2}}.

$$\frac{\left(r^{2}+s^{2}\right)r^{2}s^{2}}{rs\left(r^{4}-s^{4}\right)}$$

Cancel out rs in both numerator and denominator.

$$\frac{rs\left(r^{2}+s^{2}\right)}{r^{4}-s^{4}}$$

Factor the expressions that are not already factored.

$$\frac{rs\left(r^{2}+s^{2}\right)}{\left(r+s\right)\left(r-s\right)\left(r^{2}+s^{2}\right)}$$

Cancel out r^{2}+s^{2} in both numerator and denominator.

$$\frac{rs}{\left(r+s\right)\left(r-s\right)}$$

Expand the expression.

$$\frac{rs}{r^{2}-s^{2}}$$

Expand

$$\frac{rs}{r^{2}-s^{2}}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of s and r is rs. Multiply \frac{r}{s} times \frac{r}{r}. Multiply \frac{s}{r} times \frac{s}{s}.

$$\frac{\frac{rr}{rs}+\frac{ss}{rs}}{\frac{r^{2}}{s^{2}}-\frac{s^{2}}{r^{2}}}$$

Since \frac{rr}{rs} and \frac{ss}{rs} have the same denominator, add them by adding their numerators.

$$\frac{\frac{rr+ss}{rs}}{\frac{r^{2}}{s^{2}}-\frac{s^{2}}{r^{2}}}$$

Do the multiplications in rr+ss.

$$\frac{\frac{r^{2}+s^{2}}{rs}}{\frac{r^{2}}{s^{2}}-\frac{s^{2}}{r^{2}}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of s^{2} and r^{2} is r^{2}s^{2}. Multiply \frac{r^{2}}{s^{2}} times \frac{r^{2}}{r^{2}}. Multiply \frac{s^{2}}{r^{2}} times \frac{s^{2}}{s^{2}}.

$$\frac{\frac{r^{2}+s^{2}}{rs}}{\frac{r^{2}r^{2}}{r^{2}s^{2}}-\frac{s^{2}s^{2}}{r^{2}s^{2}}}$$

Since \frac{r^{2}r^{2}}{r^{2}s^{2}} and \frac{s^{2}s^{2}}{r^{2}s^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{r^{2}+s^{2}}{rs}}{\frac{r^{2}r^{2}-s^{2}s^{2}}{r^{2}s^{2}}}$$

Do the multiplications in r^{2}r^{2}-s^{2}s^{2}.

$$\frac{\frac{r^{2}+s^{2}}{rs}}{\frac{r^{4}-s^{4}}{r^{2}s^{2}}}$$

Divide \frac{r^{2}+s^{2}}{rs} by \frac{r^{4}-s^{4}}{r^{2}s^{2}} by multiplying \frac{r^{2}+s^{2}}{rs} by the reciprocal of \frac{r^{4}-s^{4}}{r^{2}s^{2}}.

$$\frac{\left(r^{2}+s^{2}\right)r^{2}s^{2}}{rs\left(r^{4}-s^{4}\right)}$$

Cancel out rs in both numerator and denominator.

$$\frac{rs\left(r^{2}+s^{2}\right)}{r^{4}-s^{4}}$$

Factor the expressions that are not already factored.

$$\frac{rs\left(r^{2}+s^{2}\right)}{\left(r+s\right)\left(r-s\right)\left(r^{2}+s^{2}\right)}$$

Cancel out r^{2}+s^{2} in both numerator and denominator.

$$\frac{rs}{\left(r+s\right)\left(r-s\right)}$$

Expand the expression.

$$\frac{rs}{r^{2}-s^{2}}$$