What is the value of `(1+i)^(5)-(1-i)^(5),` where `i=sqrt(-1)`?

Options

A. `-8`
B. 8
C. `8i`
D. `-8i`

Answer

A

Explanation

`(1+i)^(5)=^(5)C_(0)(1)^(5-0)i^(0)+""^(5)C_(1)(1)^(5-1)(i)^(1)+""^(5)C_(2)(1)^(5-2)(i)^(2)+""^(5)C_(3)(1)^(5-3)(i)^(3)+""^(5)C_(4)(1)^(5-5)(i)^(5)`
`=1+5i+10i^(2)+10i^(3)+5i^(4)+i^(5)....(1)`
`(1-i)^(5)=^(5)C_(0)i^(0-5)C_(1)+""^(5)C_(2)i^(2-5)C_(3)i^(3)+""^(5)C_(4)i^(4-5)C_(5)i^(5)`
`=1-5i+10i^(2) -10i^(3)+5i^(4)-i^(5)" "...(2)`
By adding (1) and (2) , we get
`(1+i)^(5)+(1-i)^(5)=1+5i+10i^(2)+10i^(3)+5i^(4)+i^(5)+1-5i+10i^(2)-10i^(3)+5i^(4)-5^(5).`
`=2+20i^(2)+10i^(4)=2-20+10=-8`