a: (x^2+y^2)^2-(2xy)^2
=(x^2+y^2-2xy)(x^2+y^2+2xy)
=(x-y)^2*(x+y)^2
b: (a+b)^3+(a-b)^3
\(=\left(a+b+a-b\right)\left[\left(a+b\right)^2-\left(a+b\right)\left(a-b\right)+\left(a-b\right)^2\right]\)
\(=2a\cdot\left[a^2+2ab+b^2-a^2+b^2+a^2-2ab+b^2\right]\)
=2a(a^2+3b^2)
c: (a+b)^3-(a-b)^3
\(=\left(a+b-a+b\right)\left[\left(a+b\right)^2+\left(a-b\right)\left(a+b\right)+\left(a-b\right)^2\right]\)
\(=2b\left(a^2+2ab+b^2+a^2-b^2+a^2-2ab+b^2\right)\)
=2b(3a^2+b^2)
d: a^6-b^6
=(a^2-b^2)(a^4+a^2b^2+b^4)
\(=\left(a^2-b^2\right)\left[a^4+2a^2b^2+b^4-a^2b^2\right]\)
\(=\left(a^2-b^2\right)\left[\left(a^2+b^2\right)^2-a^2b^2\right]\)
e: \(\left(\dfrac{x+y}{2}\right)^2+\left(\dfrac{x-y}{2}\right)^2\)
\(=\dfrac{1}{4}\left[\left(x+y\right)^2+\left(x-y\right)^2\right]\)
\(=\dfrac{1}{4}\left(2x^2+2y^2\right)=\dfrac{x^2+y^2}{2}\)
