Solve =1/r^2-1/(r-1)^2

$$= \frac { 1 } { r ^ { 2 } } - \frac { 1 } { ( r - 1 ) ^ { 2 } }$$

Evaluate

$$-\frac{2r-1}{\left(r\left(r-1\right)\right)^{2}}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of r^{2} and \left(r-1\right)^{2} is r^{2}\left(r-1\right)^{2}. Multiply \frac{1}{r^{2}} times \frac{\left(r-1\right)^{2}}{\left(r-1\right)^{2}}. Multiply \frac{1}{\left(r-1\right)^{2}} times \frac{r^{2}}{r^{2}}.

$$\frac{\left(r-1\right)^{2}}{r^{2}\left(r-1\right)^{2}}-\frac{r^{2}}{r^{2}\left(r-1\right)^{2}}$$

Since \frac{\left(r-1\right)^{2}}{r^{2}\left(r-1\right)^{2}} and \frac{r^{2}}{r^{2}\left(r-1\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(r-1\right)^{2}-r^{2}}{r^{2}\left(r-1\right)^{2}}$$

Do the multiplications in \left(r-1\right)^{2}-r^{2}.

$$\frac{r^{2}-2r+1-r^{2}}{r^{2}\left(r-1\right)^{2}}$$

Combine like terms in r^{2}-2r+1-r^{2}.

$$\frac{-2r+1}{r^{2}\left(r-1\right)^{2}}$$

Expand r^{2}\left(r-1\right)^{2}.

$$\frac{-2r+1}{r^{4}-2r^{3}+r^{2}}$$

Expand

$$\frac{1-2r}{\left(r\left(r-1\right)\right)^{2}}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of r^{2} and \left(r-1\right)^{2} is r^{2}\left(r-1\right)^{2}. Multiply \frac{1}{r^{2}} times \frac{\left(r-1\right)^{2}}{\left(r-1\right)^{2}}. Multiply \frac{1}{\left(r-1\right)^{2}} times \frac{r^{2}}{r^{2}}.

$$\frac{\left(r-1\right)^{2}}{r^{2}\left(r-1\right)^{2}}-\frac{r^{2}}{r^{2}\left(r-1\right)^{2}}$$

Since \frac{\left(r-1\right)^{2}}{r^{2}\left(r-1\right)^{2}} and \frac{r^{2}}{r^{2}\left(r-1\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(r-1\right)^{2}-r^{2}}{r^{2}\left(r-1\right)^{2}}$$

Do the multiplications in \left(r-1\right)^{2}-r^{2}.

$$\frac{r^{2}-2r+1-r^{2}}{r^{2}\left(r-1\right)^{2}}$$

Combine like terms in r^{2}-2r+1-r^{2}.

$$\frac{-2r+1}{r^{2}\left(r-1\right)^{2}}$$

Expand r^{2}\left(r-1\right)^{2}.

$$\frac{-2r+1}{r^{4}-2r^{3}+r^{2}}$$