Solve 1/u^2-1/v^2/frac{3{u}-3/v}

$$\frac { \frac { 1 } { u ^ { 2 } } - \frac { 1 } { v ^ { 2 } } } { \frac { 3 } { u } - \frac { 3 } { v } }$$

Evaluate

$$\frac{u+v}{3uv}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of u^{2} and v^{2} is u^{2}v^{2}. Multiply \frac{1}{u^{2}} times \frac{v^{2}}{v^{2}}. Multiply \frac{1}{v^{2}} times \frac{u^{2}}{u^{2}}.

$$\frac{\frac{v^{2}}{u^{2}v^{2}}-\frac{u^{2}}{u^{2}v^{2}}}{\frac{3}{u}-\frac{3}{v}}$$

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

$$\frac{\frac{v^{2}-u^{2}}{u^{2}v^{2}}}{\frac{3}{u}-\frac{3}{v}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of u and v is uv. Multiply \frac{3}{u} times \frac{v}{v}. Multiply \frac{3}{v} times \frac{u}{u}.

$$\frac{\frac{v^{2}-u^{2}}{u^{2}v^{2}}}{\frac{3v}{uv}-\frac{3u}{uv}}$$

Since \frac{3v}{uv} and \frac{3u}{uv} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{v^{2}-u^{2}}{u^{2}v^{2}}}{\frac{3v-3u}{uv}}$$

Divide \frac{v^{2}-u^{2}}{u^{2}v^{2}} by \frac{3v-3u}{uv} by multiplying \frac{v^{2}-u^{2}}{u^{2}v^{2}} by the reciprocal of \frac{3v-3u}{uv}.

$$\frac{\left(v^{2}-u^{2}\right)uv}{u^{2}v^{2}\left(3v-3u\right)}$$

Cancel out uv in both numerator and denominator.

$$\frac{-u^{2}+v^{2}}{uv\left(-3u+3v\right)}$$

Factor the expressions that are not already factored.

$$\frac{\left(u+v\right)\left(-u+v\right)}{3uv\left(-u+v\right)}$$

Cancel out -u+v in both numerator and denominator.

$$\frac{u+v}{3uv}$$

Expand

$$\frac{u+v}{3uv}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of u^{2} and v^{2} is u^{2}v^{2}. Multiply \frac{1}{u^{2}} times \frac{v^{2}}{v^{2}}. Multiply \frac{1}{v^{2}} times \frac{u^{2}}{u^{2}}.

$$\frac{\frac{v^{2}}{u^{2}v^{2}}-\frac{u^{2}}{u^{2}v^{2}}}{\frac{3}{u}-\frac{3}{v}}$$

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

$$\frac{\frac{v^{2}-u^{2}}{u^{2}v^{2}}}{\frac{3}{u}-\frac{3}{v}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of u and v is uv. Multiply \frac{3}{u} times \frac{v}{v}. Multiply \frac{3}{v} times \frac{u}{u}.

$$\frac{\frac{v^{2}-u^{2}}{u^{2}v^{2}}}{\frac{3v}{uv}-\frac{3u}{uv}}$$

Since \frac{3v}{uv} and \frac{3u}{uv} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{v^{2}-u^{2}}{u^{2}v^{2}}}{\frac{3v-3u}{uv}}$$

Divide \frac{v^{2}-u^{2}}{u^{2}v^{2}} by \frac{3v-3u}{uv} by multiplying \frac{v^{2}-u^{2}}{u^{2}v^{2}} by the reciprocal of \frac{3v-3u}{uv}.

$$\frac{\left(v^{2}-u^{2}\right)uv}{u^{2}v^{2}\left(3v-3u\right)}$$

Cancel out uv in both numerator and denominator.

$$\frac{-u^{2}+v^{2}}{uv\left(-3u+3v\right)}$$

Factor the expressions that are not already factored.

$$\frac{\left(u+v\right)\left(-u+v\right)}{3uv\left(-u+v\right)}$$

Cancel out -u+v in both numerator and denominator.

$$\frac{u+v}{3uv}$$