Solve 2r/r^2-s^2-4rs/(r+s)^2(r-s)-r-s/(r+s)^2

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

Evaluate

$$\frac{1}{r+s}$$

Short Solution Steps

Factor r^{2}-s^{2}.

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

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

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

Since \frac{2r\left(r+s\right)}{\left(r-s\right)\left(r+s\right)^{2}} and \frac{4rs}{\left(r-s\right)\left(r+s\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 2r\left(r+s\right)-4rs.

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

Combine like terms in 2r^{2}+2rs-4rs.

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

Factor the expressions that are not already factored in \frac{2r^{2}-2rs}{\left(r-s\right)\left(r+s\right)^{2}}.

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

Cancel out r-s in both numerator and denominator.

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

Since \frac{2r}{\left(r+s\right)^{2}} and \frac{r-s}{\left(r+s\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 2r-\left(r-s\right).

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

Combine like terms in 2r-r+s.

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

Cancel out r+s in both numerator and denominator.

$$\frac{1}{r+s}$$

Expand

$$\frac{1}{r+s}$$

Short Solution Steps

Factor r^{2}-s^{2}.

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

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

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

Since \frac{2r\left(r+s\right)}{\left(r-s\right)\left(r+s\right)^{2}} and \frac{4rs}{\left(r-s\right)\left(r+s\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 2r\left(r+s\right)-4rs.

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

Combine like terms in 2r^{2}+2rs-4rs.

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

Factor the expressions that are not already factored in \frac{2r^{2}-2rs}{\left(r-s\right)\left(r+s\right)^{2}}.

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

Cancel out r-s in both numerator and denominator.

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

Since \frac{2r}{\left(r+s\right)^{2}} and \frac{r-s}{\left(r+s\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 2r-\left(r-s\right).

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

Combine like terms in 2r-r+s.

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

Cancel out r+s in both numerator and denominator.

$$\frac{1}{r+s}$$