Use the Binomial Theorem to expand the complex number. Simplify your result.

Question

Use the Binomial Theorem to expand the complex number. Simplify your result.

(5 - √ 3i)^4

Answer 1

∑_n=0⁴ C(4,n)5^n (-√3 i)^(4-n) = C(4,0) (-√3 i)^4 + C(4,1)5^1 (-√3 i)^3 + C(4,2)5^2 (-√3 i)^2 + C(4,1)5^3 (-√3 i)^1 + C(4,0)5^4 =

9i^4 + 4*5 * -3√3i^3 + 6 * 25 * 3i^2 + 4 * 125 * (-√3 i) + 625 = (as i^2 = -1 and i^4 = 1)

9 + 60√3 i - 450 - 500√3 i + 625 =

184 - 440√3 i

Note: it is a lot easier to solve this problem using DeMoivre's Theorem, recognizing that you will need to divide the imaginary component by sqrt(3) to put it into this form. I actually checked my math this way

5 - sqrt(3) i has magnitude sqrt(5^2 + sqrt(3)^2) = sqrt(28) and angle atan(-sqrt(3)/5), so

(5-sqrt(3)i)^4 has magnitude ( sqrt(28))^4 = 784 and direction 4 atan(-sqrt(3)/5), which you can verify is the case for our answer above, 184 - 440√3 i.