a) \(\left(3+1\right)\left(3^2+1\right)\left(3^4+1\right)\left(3^8+1\right)\left(3^{16}+1\right)\left(3^{32}+1\right)\)

\(=\dfrac{1}{2}\left(3-1\right)\left(3+1\right)\left(3^2+1\right)\left(3^4+1\right)\left(3^8+1\right)\left(3^{16}+1\right)\left(3^{32}+1\right)\)

\(=\dfrac{1}{2}\left(3^2-1\right)\left(3^2+1\right)\left(3^4+1\right)\left(3^8+1\right)\left(3^{16}+1\right)\left(3^{32}+1\right)\)

\(=\dfrac{1}{2}\left(3^4-1\right)\left(3^4+1\right)\left(3^8+1\right)\left(3^{16}+1\right)\left(3^{32}+1\right)\)

\(=\dfrac{1}{2}\left(3^8-1\right)\left(3^8+1\right)\left(3^{16}+1\right)\left(3^{32}+1\right)\)

\(=\dfrac{1}{2}\left(3^{16}-1\right)\left(3^{16}+1\right)\left(3^{32}+1\right)\)

\(=\dfrac{1}{2}\left(3^{32}-1\right)\left(3^{32}+1\right)\)

\(=\dfrac{1}{2}\left(3^{64}-1\right)\)

\(=\dfrac{3^{64}-1}{2}\)

b) \(\left(a+b+c\right)2+\left(a-b-c\right)2+\left(b-c-a\right)2+\left(c-a-b\right)2\)

\(=2\left[\left(a+b+c\right)+\left(a-b-c\right)+\left(b-c-a\right)+\left(c-a-b\right)\right]\)

\(=2\left(a+b+c+a-b-c+b-c-a+c-a-b\right)\)

\(=2.0\)

\(=0\)

c)\(\left(a+b+c+d\right)2+\left(a+b-c-d\right)2+\left(a+c-b-d\right)2+\left(a+d-b-c\right)2\)

\(=2\left(a+b+c+d+a+b-c-d+a+c-b-d+a+d-b-c\right)\)

\(=2.4a\)

\(=8a\)