Question

Answer 1

a: \(76^2-24^2=\left(76-24\right)\left(76+24\right)\)

\(=52\cdot100=5200\)

b: \(87^2-13^2=\left(87-13\right)\left(87+13\right)=74\cdot100=7400\)

c: \(82\cdot78=\left(80-2\right)\left(80+2\right)=80^2-4=6400-4=6396\)

d: \(108\cdot92=\left(100+8\right)\left(100-8\right)=100^2-8^2\)

=10000-64

=9936

e: \(15\left(4^2+1\right)\left(4^4+1\right)\left(4^8+1\right)\left(4^{16}+1\right)-\left(4^{32}-11\right)\)

\(=\left(4^2-1\right)\left(4^2+1\right)\left(4^4+1\right)\left(4^8+1\right)\left(4^{16}+1\right)-\left(4^{32}-11\right)\)

\(=\left(4^4-1\right)\left(4^4+1\right)\left(4^8+1\right)\left(4^{16}+1\right)-\left(4^{32}-11\right)\)

\(=\left(4^8-1\right)\left(4^8+1\right)\left(4^{16}+1\right)-\left(4^{32}-11\right)\)

\(=\left(4^{16}-1\right)\left(4^{16}+1\right)-\left(4^{32}-11\right)\)

\(=4^{32}-1-4^{32}+11=10\)