Question

How do you use the Binomial theorem to expand `(4-5i)^3`?

Answer 1

`(4-5i)^3 = -236-115i`

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=4,b=-5i,n=3` We know, `nC_r = (n!)/(r!*(n-r)!`
`:.3C_0 =1 , 3C_1 =3, 3C_2 =3,3C_3 =1 ; i^2=-1 ,i ^3 = -i`

`:.(4-5i)^3 = 4^3+3*4^2*(-5i) +3*4*(-5i)^2+(-5i)^3` or

`(4-5i)^3 = 64-240i+300i^2-125i^3` or

`(4-5i)^3 = 64-240i-300+125i` or

`(4-5i)^3 = -236-115i` [Ans]