Video: Finding the Powers of Complex Numbers

Video Transcript

What is one plus 𝑖 to the power of 10?

Now what we don’t want to do here is to expand these brackets using a long handed or binomial method. In fact, we’re going to recall De Moivre’s theorem. Now, De Moivre’s theorem works for numbers both in exponential and trigonometric form. Let’s look at the exponential form.

For 𝑧 is equal to 𝑟𝑒 to the 𝑖𝜃, 𝑧 to the power of 𝑛 is equal to 𝑟 to the power of 𝑛 𝑒 to the 𝑖𝑛𝜃. So if we can find a way to convert our complex number, one plus 𝑖, into exponential form, we can apply De Moivre’s theorem to work out what one plus 𝑖 to the power of 10 is.

And we can convert a complex number of the form 𝑎 plus 𝑏𝑖 into exponential form using the formula 𝑟𝑒 to the 𝑖𝜃, where 𝑟, the modulus, is the square root of 𝑎 squared plus 𝑏 squared. And 𝜃, the argument, is arctan or inverse tan of 𝑏 divided by 𝑎. For our complex number, one plus 𝑖, 𝑎 is the constant. It’s one. And 𝑏 is the coefficient of 𝑖. It’s also one.

And we can find the modulus then of our complex number by finding the square root of 𝑎 squared plus 𝑏 squared, which is the square root of one squared plus one squared. So the modulus is root two. And the argument is the inverse tan or arctan of one divided by one, which is 𝜋 by four. So we can say that our complex number, one plus 𝑖, is the same as root two 𝑒 to the 𝜋 by four 𝑖.

And now, we apply De Moivre’s theorem directly to this complex number. The modulus will be root two to the power of 10. And the argument will be found by multiplying 𝜋 by four by 10. Now, if we think about root two as two to the power of half, we can say that root two to the power of 10 is two to the power of a half to the power of 10. And we know we need to multiply these powers. So this is two to the power of five, which is equal to 32. And 10 multiplied by 𝜋 by four is 10𝜋 by four or five 𝜋 by two.

So we now know the magnitude or the modulus and the argument of our complex number, one plus 𝑖 to the power of 10. So how do we convert this back into rectangular form? Well, this comes from the trigonometric form of a complex number. And we use these two formulae. 𝑎 is equal to 𝑟 cos 𝜃. And 𝑏 is equal to 𝑟 sin 𝜃.

In this case, 𝑎 is equal to 32 cos of five 𝜋 by two. And 𝑏 is equal to 32 sin of five 𝜋 by two. Cos of five 𝜋 by two is zero. So 𝑎 is equal to zero. And sin of five 𝜋 by two is one. So 𝑏 is equal to 32.

And since we saw earlier that 𝑏 is the coefficient of 𝑖, we can say that one plus 𝑖 to the power of 10 is the same as 32𝑖.