a: \(\left(a+b\right)^3+\left(a-b\right)^3\)
\(=a^3+3a^2b+3ab^2+b^3+a^3-3a^2b+3ab^2-b^3\)
\(=2a^3+6ab^2\)
b: \(\left(x-y\right)^3-3\left(y-x\right)^2z+3\left(x-y\right)\cdot z^2-z^3\)
\(=\left(x-y\right)^3-3\left(x-y\right)^2z+3\left(x-y\right)\cdot z^2-z^3\)
\(=\left(x-y-z\right)^3\)
c: \(6\left(c-d\right)\left(c+d\right)^2+12\left(c-d\right)^2\cdot\left(c+d\right)+\left(c+d\right)^3+8\left(c-d\right)^3\)
\(=\left(c+d\right)^3+6\cdot\left(c+d\right)^2\cdot\left(c-d\right)+12\cdot\left(c+d\right)\left(c-d\right)^2+8\left(c-d\right)^3\)
\(=\left(c+d+2c-2d\right)^3=\left(3c-d\right)^3\)
d: \(\left(m-n\right)^3-\left(n+p\right)^3-3\left(n+p\right)^2\cdot\left(n-m\right)-3\left(n+p\right)\left(n-m\right)^2\)
\(=\left(m-n\right)^3-3\left(m-n\right)^2\cdot\left(n+p\right)+3\left(m-n\right)\left(n+p\right)^2-\left(n+p\right)^3\)
=(m-n-n-p)^3=(m-2n-p)^3
e: \(\left(u-v\right)^3+3uv\left(u+v\right)\)
\(=u^3-3u^2v+3uv^2-v^3+3u^2v+3uv^2=u^3-v^3+6uv^2\)
f: \(3\left(c-2d\right)\left(c+2d\right)^2+3\left(c-2d\right)^2\cdot\left(c+2d\right)+\left(c+2d\right)^3+\left(c-2d\right)^3\)
\(=\left(c+2d\right)^3+3\cdot\left(c+2d\right)^2\cdot\left(c-2d\right)+3\cdot\left(c+2d\right)\left(c-2d\right)^2+\left(c-2d\right)^3\)
=(c+2d+c-2d)^3
=(2c)^3=8c^3
