Question

How do you use the binomial theorem to expand `(1+i)^4`?

Answer 1

`(1+i)^4 = -4`

Explanation:

We know `(a+b)^n= nC_0 a^n*b^0 +nC_1 a^(n-1)*b^1 + nC_2 a^(n-2)*b^2+..........+nC_n a^(n-n)*b^n`

Here `a=1,b=i,n=4` We know, `nC_r = (n!)/(r!*(n-r)!`
`:.4C_0 =1 , 4C_1 =4, 4C_2 =6,4C_3 =4, 4C_4 =1`
`i^2=-1 ,i^3= -i,i^4=1`

`:.(1+i)^4 = 1^4+4*1^3*i+6*1^2*i^2+4*1*i^3+i^4` or

`(1+i)^4 = 1+4*i+6*i^2+4*i^3+i^4` or

`(1+i)^4 = 1+cancel(4i)-6-cancel(4i)+1 = -4` [Ans]