By this point we are really good at solving equations and even raising numbers to powers, right?!
But, if our numbers are complex that makes finding its power a little more challenging. Moreover, trying to find all roots or solutions to an equations when we a fairly certain the answers contain complex numbers is even more difficult.
Thankfully, we have De Moivre’s Theorem, and its extension, the Complex Root Theorem.
Finding Powers is super easy as long as our Complex Number is first converted from Standard Form to Polar Form. Because in doing so, all we have to do is apply De Moivre’s Theorem, which is sometimes referred to as De Moivre’s Formula or Identity as Wikipedia nicely points out.
With this formula at our fingertips we will quickly and easily be able to find the powers of any complex number, even when the exponent is negative!
De Moivre’s Theorem
Now, in that same vein, if we can raise a complex number to a power, we should be able to find all of its roots too.
We know from the Fundamental Theorem of Algebra, that every nonzero number has exactly n-distinct roots. Hence, the Complex Root Theorem, or nth Root Theorem.
Again, we will first have to ensure that our Complex Number is in Polar Form, and then all we have to do is apply the formula. Simple!
Furthermore, with the Complex Roots Theorem, we can solve any equation. As CK-12 points out, the roots of a complex number are cyclic in nature, which means the nth roots are equally spaced on the circumference of a circle.
Huh?
In other words, they come in pairs – conjugate pairs!
Cool!
And to top it all off, this lesson proves that you are smarter than your calculator! Yeah!
De Moivre’s Theorem – Video
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