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\left(\frac{1}{\left(m+n\right)^{2}}\left(\frac{n^{2}}{m^{2}n^{2}}+\frac{m^{2}}{m^{2}n^{2}}\right)+\frac{2}{\left(m+n\right)^{3}}\left(\frac{1}{m}+\frac{1}{n}\right)\right)m^{2}n^{2}
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of m^{2} and n^{2} is m^{2}n^{2}. Multiply \frac{1}{m^{2}} times \frac{n^{2}}{n^{2}}. Multiply \frac{1}{n^{2}} times \frac{m^{2}}{m^{2}}.
\left(\frac{1}{\left(m+n\right)^{2}}\times \frac{n^{2}+m^{2}}{m^{2}n^{2}}+\frac{2}{\left(m+n\right)^{3}}\left(\frac{1}{m}+\frac{1}{n}\right)\right)m^{2}n^{2}
Since \frac{n^{2}}{m^{2}n^{2}} and \frac{m^{2}}{m^{2}n^{2}} have the same denominator, add them by adding their numerators.
\left(\frac{n^{2}+m^{2}}{\left(m+n\right)^{2}m^{2}n^{2}}+\frac{2}{\left(m+n\right)^{3}}\left(\frac{1}{m}+\frac{1}{n}\right)\right)m^{2}n^{2}
Multiply \frac{1}{\left(m+n\right)^{2}} times \frac{n^{2}+m^{2}}{m^{2}n^{2}} by multiplying numerator times numerator and denominator times denominator.
\left(\frac{n^{2}+m^{2}}{\left(m+n\right)^{2}m^{2}n^{2}}+\frac{2}{\left(m+n\right)^{3}}\left(\frac{n}{mn}+\frac{m}{mn}\right)\right)m^{2}n^{2}
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of m and n is mn. Multiply \frac{1}{m} times \frac{n}{n}. Multiply \frac{1}{n} times \frac{m}{m}.
\left(\frac{n^{2}+m^{2}}{\left(m+n\right)^{2}m^{2}n^{2}}+\frac{2}{\left(m+n\right)^{3}}\times \frac{n+m}{mn}\right)m^{2}n^{2}
Since \frac{n}{mn} and \frac{m}{mn} have the same denominator, add them by adding their numerators.
\left(\frac{n^{2}+m^{2}}{\left(m+n\right)^{2}m^{2}n^{2}}+\frac{2\left(n+m\right)}{\left(m+n\right)^{3}mn}\right)m^{2}n^{2}
Multiply \frac{2}{\left(m+n\right)^{3}} times \frac{n+m}{mn} by multiplying numerator times numerator and denominator times denominator.
\left(\frac{n^{2}+m^{2}}{\left(m+n\right)^{2}m^{2}n^{2}}+\frac{2}{mn\left(m+n\right)^{2}}\right)m^{2}n^{2}
Cancel out m+n in both numerator and denominator.
\left(\frac{n^{2}+m^{2}}{m^{2}n^{2}\left(m+n\right)^{2}}+\frac{2mn}{m^{2}n^{2}\left(m+n\right)^{2}}\right)m^{2}n^{2}
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(m+n\right)^{2}m^{2}n^{2} and mn\left(m+n\right)^{2} is m^{2}n^{2}\left(m+n\right)^{2}. Multiply \frac{2}{mn\left(m+n\right)^{2}} times \frac{mn}{mn}.
\frac{n^{2}+m^{2}+2mn}{m^{2}n^{2}\left(m+n\right)^{2}}m^{2}n^{2}
Since \frac{n^{2}+m^{2}}{m^{2}n^{2}\left(m+n\right)^{2}} and \frac{2mn}{m^{2}n^{2}\left(m+n\right)^{2}} have the same denominator, add them by adding their numerators.
\frac{\left(m+n\right)^{2}}{m^{2}n^{2}\left(m+n\right)^{2}}m^{2}n^{2}
Factor the expressions that are not already factored in \frac{n^{2}+m^{2}+2mn}{m^{2}n^{2}\left(m+n\right)^{2}}.
\frac{1}{m^{2}n^{2}}m^{2}n^{2}
Cancel out \left(m+n\right)^{2} in both numerator and denominator.
\frac{m^{2}}{m^{2}n^{2}}n^{2}
Express \frac{1}{m^{2}n^{2}}m^{2} as a single fraction.
\frac{1}{n^{2}}n^{2}
Cancel out m^{2} in both numerator and denominator.
1
Cancel out n^{2} and n^{2}.