Question

If `(a + i b) (c + i d) (e + i f) (g + i h) = A + i B`, then show that `(a^2+b^2)(c^2+d^2)(e^2+f^2)(g^2+h^2)=A^2+B^2`

Answer 1

Given :
`(a+ib)(c+i d)(e+ i f)(g+ih)=A+iB`
Taking modulus of both sides
`" "|(a+ib)(c+i d)(e+i f)(g+ih)|=|A+ iB|`
`rArr" " |a+ ib||c+i d||e+ i f||g+ih|=|A+iB|`
`rArr" "sqrt( a^(2)+b^(2))*sqrt(c^(2)+d^(2))sqrt(e^(2)+f^(2))sqrt(g^(2)+h^(2)) =sqrt(A^(2)+B^(2))`
Squaring both sides,
`(a^(2)+b^(2))(c^(2)+d^(2)(e^(2)+f^(2))(g^(2)+h^(2))=(A^(2)+B^(2))`