# How do I find (3+i)^4?

Sep 16, 2014

I like to use Pascal's Triangle to do binomial expansions!

The triangle helps us to find the coefficients of our "expansion" so that we don't have to do the Distributive property so many times! (it actually represents how many of the like terms we have gathered)

So, in the form ${\left(a + b\right)}^{4}$ we use the row: 1, 4, 6, 4, 1.

$1 {\left(a\right)}^{4} + 4 {\left(a\right)}^{3} \left(b\right) + 6 {\left(a\right)}^{2} {\left(b\right)}^{2} + 4 \left(a\right) {\left(b\right)}^{3} + {\left(b\right)}^{4}$

But your example contains a = 3 and b = i. So...

$1 {\left(3\right)}^{4} + 4 {\left(3\right)}^{3} \left(i\right) + 6 {\left(3\right)}^{2} {\left(i\right)}^{2} + 4 \left(3\right) {\left(i\right)}^{3} + {\left(i\right)}^{4}$

$= 81 + 4 \left(27 i\right) + 6 \left(9 {i}^{2}\right) + 12 \left({i}^{3}\right) + 1$

$= 81 + 108 i - 54 - 12 i + 1$

$= 28 + 96 i$