`(5 - sqrt(3) i )^4` Use the Binomial Theorem to expand the complex number, then simplify the result.

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 `