# Can a complex number be written Cartesian form in terms of i to a power other than 1? And if so...

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I came across the question:

#z_1 = -2 -2sqrt(3i)#

#z_2 = 3sqrt(3) + 3i#

Find #z_3 = z_1 z_2# in Cartesian form.

But I keep ending up with multiple coefficients in terms of i, but each to a different power, such as

#-6sqrt(3) + 6isqrt(3i) -18sqrt(i) -6i# .

How can this be written in Cartesian form properly, or did I do something wrong to reach this?

I came across the question:

Find

But I keep ending up with multiple coefficients in terms of i, but each to a different power, such as

How can this be written in Cartesian form properly, or did I do something wrong to reach this?

##### 1 Answer

The square roots of

#### Explanation:

Yes, this number can be written in Cartesian form. The problem at the moment is that you have a

Your problem seems to be with taking the square root of a complex number. There are two ways to do this: one with just knowledge of basic algebra and one (far more useful) with knowledge of Euler's identity.

**Way 1: Basic algebra.**

Let

Then,

Equating coeffecients

Then

We can check this by squaring

**Way 2: Eulers identity**

Let

The sine and cosine functions have periodicity

Then,

So, for this specific example, let

The number

Then we conclude

Then choose

When we raise to a power

I hope you can see how this method is more applicable to finding the roots of complex numbers more generally (and demonstrates that they can always be written in Cartesian form).

I trust you can use