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Algebraic identity for (a+b)^2

Simplify the ...

Question

Simplify the expression:
$\frac{(m + n)^{2} + (m^{2} - n^{2})}{2 m^{5} - 2 m n^{4}}$

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Solution

Given expression: $\frac{(m + n)^{2} + (m^{2} - n^{2})}{2 m^{5} - 2 m n^{4}}$

$\frac{(m + n)^{2} + (m^{2} - n^{2})}{2 m^{5} - 2 m n^{4}} \Rightarrow \frac{(m + n)^{2} + (m^{2} - n^{2})}{(2 m) (m^{4} - n^{4})}$

By using the identities,
$(a + b)^{2} = a^{2} + 2 a b + b^{2} (a^{2} - b^{2}) = (a + b) (a - b)$

$\Rightarrow \frac{(m^{2} + n^{2} + 2 m n + m^{2} - n^{2})}{2 m (m^{2} + n^{2}) (m^{2} - n^{2})} \Rightarrow \frac{2 m^{2} + 2 m n}{2 m (m^{2} + n^{2}) (m^{2} - n^{2})} \Rightarrow \frac{2 m (m + n)}{2 m (m^{2} + n^{2}) (m + n) (m - n)} \Rightarrow \frac{1}{(m + n) (m^{2} + n^{2})}$

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

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Q. Simplify by using formula

M^{2} - N^{2}

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