If 𝑧 one is equal to cos of 210 degrees plus 𝑖 sin of 210 degrees, 𝑧 two is equal to three cos of 135 degrees plus 𝑖 sin of 135 degrees, and 𝑧 three is equal to four cos of 135 degrees plus 𝑖 sin of 135 degrees, what is the exponential form of 𝑧 one multiplied by 𝑧 two multiplied by 𝑧 three all to the power of four.
We have been given three complex numbers in polar, or trigonometric form. We’re being asked to find their product, and then we need to raise that to the power of four. And we’re also being asked to give our answer in exponential form. There are two ways we can go about this. We could begin by finding their product in polar form. We could then raise that to the power of four, and then convert it to exponential form.
Alternatively, we could convert each complex number into exponential form first. We could then find their product, and then raise that number to the power of four. In fact, we’re going to apply DeMoivre’s theorem no matter which method we use. So, let’s look at finding their product in polar form first.
To find the product of complex numbers, it’s fairly straightforward. We multiply their moduli, and we add their arguments. Now I’ve written this list of complex numbers in polar form, but this also works for complex numbers in exponential form. Let’s begin by multiplying the moduli of these three complex numbers.
The modulus of our first complex number is one. The modulus of our second complex number is three. And the modulus of our third complex number is four. So, the modulus of the product of these three complex numbers is one multiplied by three multiplied by four, which is, of course, 12.
Next, we’re going to add their arguments. Now we generally like to represent our arguments in radians, but actually it’s easier to add them in degrees and convert them in a moment. The argument of our first complex number is 210. For our second complex number, it’s 135. And for our third complex number, it’s also 135. And so, the argument for the product of our complex numbers is going to be 210 plus 135 plus 135, which is 480 degrees.
Now it doesn’t really matter if we carry on working with degrees at this stage. But we may as well turn four 480 degrees into radians, as we’re going to need to do that represent it in exponential form. Here, we need to recall that two 𝜋 radians is equal to 360 degrees. And if we divide through by 360, we see that one degree is equal to two 𝜋 by 360 radians, or 𝜋 by 180 radians. And this means we can convert from degrees to radians by multiplying by 𝜋 by 180. 480 multiplied by 𝜋 by 180 is eight 𝜋 by three. So, we can say that the argument of the product of these three complex numbers is eight 𝜋 by three radians.
And we can now represent the product of these three complex numbers in polar form. It’s 12 multiplied by cos of eight 𝜋 by three plus 𝑖 sin of eight 𝜋 by three. Next, we need to raise the sum to the power of four. Again, we can do this before or after changing it into exponential form. Either way we’re going to use DeMoivre’s theorem, so it doesn’t really matter.
DeMoivre’s theorem says that for a complex number 𝑧 in the form 𝑟 cos 𝜃 plus 𝑖 sin 𝜃, 𝑧 to the power of 𝑛 is equal to 𝑟 to the power of 𝑛 multiplied by cos of 𝑛𝜃 plus 𝑖 sin of 𝑛𝜃. And that’s when 𝑛 is a natural number. So, this means we can find the modulus of 𝑧 one 𝑧 two 𝑧 three to the power of four by finding 12 to the power of four, which is 20736.
And we need to multiply the argument by that power; it’s four. So, the argument of 𝑧 one 𝑧 two 𝑧 three to the power of four is four multiplied by eight 𝜋 by three. Four multiplied by eight 𝜋 by three is 32𝜋 by three. So, at this stage, we can see that our complex number is 20736 multiplied by cos of 32𝜋 by three plus 𝑖 sin of 32𝜋 by three. So, how do we represent this in exponential form?
Well, for a complex number 𝑧 in polar form 𝑟 cos 𝜃 plus 𝑖 sin 𝜃, we can represent in exponential form as 𝑟𝑒 to the 𝑖𝜃. Now it’s important at this stage that the argument is in radians. And we’ve already done that conversion, so we’re good to go.
We can take the modulus of 20736 and the argument of 32𝜋 by three and substitute it into that formula. And we see that our complex number is 20736𝑒 to the power of 32𝜋 by three 𝑖. And there’s one thing left to do. We prefer to write our numbers in terms of their principal argument. That’s greater than negative 𝜋 and less than or equal to 𝜋.
We can see the 32𝜋 by three is considerably larger than 𝜋. And we achieve the principal argument by adding or subtracting multiples of two 𝜋. In this case, we’re going to have to subtract five lots of two 𝜋. And when we do, we get two 𝜋 by three, or two-thirds 𝜋 as our principal argument. And now we really are done. We can say that the exponential form of 𝑧 one 𝑧 two 𝑧 three to the power of four is 20736𝑒 to the power of two 𝜋 by three 𝑖.