Question

\(\left(\frac{5x+y}{x^2-5xy}+\frac{5x-y}{x^2+5xy}\right).\frac{x^2-25y^2}{x^2+y^2}\)

Answer 1

\(\left(\frac{5x+y}{x^2-5xy}+\frac{5x-y}{x^2+5xy}\right).\frac{x^2-25y^2}{x^2+y^2}\)

\(=\left(\frac{5x+y}{x\left(x-5y\right)}+\frac{5x-y}{x\left(x+5y\right)}\right).\frac{\left(x-5y\right)\left(x+5y\right)}{x^2+y^2}\)

\(=\frac{\left(5x+y\right)\left(x+5y\right)+\left(5x-y\right)\left(x-5y\right)}{x\left(x-5y\right)\left(x+5y\right)}.\frac{\left(x-5y\right)\left(x+5y\right)}{x^2+y^2}\)

\(=\frac{10\left(x^2+y^2\right)}{x\left(x^2+y^2\right)}=\frac{10}{x}\)

Answer 2

\(\left(\frac{5x+y}{x^2-5xy}+\frac{5x-y}{x^2+5xy}\right).\frac{x^2-25y^2}{x^2+y^2}\)

\(=\left(\frac{5x+y}{x\left(x-5y\right)}+\frac{5x-y}{x\left(x+5y\right)}\right)\frac{\left(x-5y\right)\left(x+5y\right)}{x^2+y^2}\)

\(=\frac{\left(5x+y\right)\left(x+5y\right)+\left(5x-y\right)\left(x-5y\right)}{x\left(x-5y\left(x+4y\right)\right)}.\frac{\left(x-5y\right)\left(x+5y\right)}{x^2+y^2}\)

\(=\frac{10\left(x^2+y^2\right)}{x\left(x^2+y^2\right)}=\frac{10}{x}\)