Solve frac{m^{2}-n^{2}}{(m+n)^{2}}*m^2+mn+n^2/m^3-n^3

$$\frac { m ^ { 2 } - n ^ { 2 } } { ( m + n ) ^ { 2 } } \times \frac { m ^ { 2 } + m n + n ^ { 2 } } { m ^ { 3 } - n ^ { 3 } }$$

Evaluate

$$\frac{1}{m+n}$$

Short Solution Steps

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

$$\frac{\left(m+n\right)\left(m-n\right)}{\left(m+n\right)^{2}}\times \frac{m^{2}+mn+n^{2}}{m^{3}-n^{3}}$$

Cancel out m+n in both numerator and denominator.

$$\frac{m-n}{m+n}\times \frac{m^{2}+mn+n^{2}}{m^{3}-n^{3}}$$

Factor the expressions that are not already factored in \frac{m^{2}+mn+n^{2}}{m^{3}-n^{3}}.

$$\frac{m-n}{m+n}\times \frac{m^{2}+mn+n^{2}}{\left(m-n\right)\left(m^{2}+mn+n^{2}\right)}$$

Cancel out m^{2}+mn+n^{2} in both numerator and denominator.

$$\frac{m-n}{m+n}\times \frac{1}{m-n}$$

Multiply \frac{m-n}{m+n} times \frac{1}{m-n} by multiplying numerator times numerator and denominator times denominator.

$$\frac{m-n}{\left(m+n\right)\left(m-n\right)}$$

Cancel out m-n in both numerator and denominator.

$$\frac{1}{m+n}$$

Expand

$$\frac{1}{m+n}$$

Short Solution Steps

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

$$\frac{\left(m+n\right)\left(m-n\right)}{\left(m+n\right)^{2}}\times \frac{m^{2}+mn+n^{2}}{m^{3}-n^{3}}$$

Cancel out m+n in both numerator and denominator.

$$\frac{m-n}{m+n}\times \frac{m^{2}+mn+n^{2}}{m^{3}-n^{3}}$$

Factor the expressions that are not already factored in \frac{m^{2}+mn+n^{2}}{m^{3}-n^{3}}.

$$\frac{m-n}{m+n}\times \frac{m^{2}+mn+n^{2}}{\left(m-n\right)\left(m^{2}+mn+n^{2}\right)}$$

Cancel out m^{2}+mn+n^{2} in both numerator and denominator.

$$\frac{m-n}{m+n}\times \frac{1}{m-n}$$

Multiply \frac{m-n}{m+n} times \frac{1}{m-n} by multiplying numerator times numerator and denominator times denominator.

$$\frac{m-n}{\left(m+n\right)\left(m-n\right)}$$

Cancel out m-n in both numerator and denominator.

$$\frac{1}{m+n}$$