Precalculus, Chapter 9, 9.5, Section 9.5, Problem 78

You need to use the binomial formula, such that:
(x+y)^n = sum_(k=0)^n ((n),(k)) x^(n-k) y^k
You need to replace 5 for x, sqrt3*i for y and 4 for n, such that:
(5 - sqrt3*i)^4 = 4C0 (5)^4+4C1 (5)^3*(-sqrt3*i)^1+4C2 (5)^2*(-sqrt3*i)^2+4C3 (5)^1*(-sqrt3*i)^3 + 4C4 3a*(-sqrt3*i)^4
By definition, nC0 = nCn = 1, hence 4C0 = 4C4 = 1.
By definition nC1 = nC(n-1) = n, hence 4C1 = 4C3 = 4.
By definition nC2 = (n(n-1))/2, hence 4C2 = 6.
(5 - sqrt3*i)^4 = 625 - 500sqrt3*i+450*i^2- 60sqrt3*i^3 + 9*i^4
Using the powers of i yields:
i = i; i^2 = -1, i^3 = -i, i^4 = 1
(5 - sqrt3*i)^4 = 184 - 440sqrt3*i
Hence, expanding the complex number using binomial theorem yields the simplified result (5 - sqrt3*i)^4 = 184 - 440sqrt3*i