`(1 + i)^4` Use the Binomial Theorem to expand the complex number, then simplify the result.

We are required to expand a binomial that contains a complex number. In order to do this we shall use the binomial theorem. This is used as follows: 

where: 

a = first term

b = last term 

n = exponent (power raised to of the binomial) 

k = term number - 1

So let's use the above equation to determine the sum of our binomial:

`(1+i)^4 =sum_(k=1)^4 ((4!)/((4-k)!*k!)) *a^(4-k)*b^k` 

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`(1+i)^4 = ((4!)/ ((4-0)!*0!)) * (1^(4-0)) * (i^0) + ((4!)/((4-1)!1!)) * 1^(4-1) * (i^1) + ((4!)/ ((4-2)! *2!)) *1^(4-2) (i^2) + ((4!)/((4-3)! *3!)) * 1^(4-3) *( i^3) + ((4!)/((4-4)!*4!)) * a^(4-4) *(i^4)`

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

Let's revise our complex rules: 

`i^ 1 = i, i^2 =-1, i^3 = -i, i^4 = 1`

`(1+i)^4 = 1 + 4i + 6(-1) + 4(-i) + 1 = -4`

ANSWER: -4

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