Question Video: Evaluating Powers of a Complex Number Mathematics • 12th Grade

Video Transcript

What is one minus two 𝑖 to the power four?

The meaning of the fourth power of a complex number is exactly the same as it would be for a real number. You just find the product of four copies of that number, so one minus two 𝑖 to the power four is one minus two 𝑖 times one minus two 𝑖 times one minus two 𝑖 times one minus two 𝑖.

Let’s work up to this by finding one minus two 𝑖 to the power of two, which is one minus two 𝑖 squared, which is one minus two 𝑖 times one minus two 𝑖. We expand the right-hand side using the distributive property just as we would if we had an 𝑥 instead of 𝑖 here. The only difference being that as 𝑖 squared is equal to negative one, whenever we see and 𝑖 squared, we can replace that with a negative one in order to simplify.

So the plus four 𝑖 squared becomes minus four. And combining our two constant terms, we find that one minus two 𝑖 squared equals negative three minus four 𝑖. Now that we found one minus two 𝑖 squared, we can move on to one minus two 𝑖 cubed. And the exponent laws that we know and love still hold for complex numbers, and so one minus two 𝑖 cubed is one minus two 𝑖 squared times one minus two 𝑖.

And we can substitute in the value of one minus two 𝑖 squared we got earlier, and we can expand this out. We get negative three plus six 𝑖 minus four 𝑖 plus eight 𝑖 squared. And again because 𝑖 squared is negative one, eight 𝑖 squared is negative eight. Combining this with the other real term, the negative three, we get negative 11.

And we can also combine the imaginary terms to get us plus two 𝑖. So one minus two 𝑖 cubed equals negative 11 plus two 𝑖. Now we’re ready to tackle the question what is one minus two 𝑖 to the power four. Well it’s one minus two 𝑖 cubed times one minus two 𝑖, which is negative 11, plus two 𝑖 times one minus two 𝑖.

Expanding that we get negative 11 plus 22𝑖 plus two 𝑖 plus four 𝑖 squared. And remembering that 𝑖 squared is negative one and combining the real and imaginary terms, we get our final answer: negative seven plus 24𝑖. That was one method to show that one minus two 𝑖 to the power four is negative seven plus 24𝑖.

Another way would be to use binomial expansion. In this case, we’d have to not only use the fact that 𝑖 squared is equal to negative one, but also the fact that as 𝑖 cubed is 𝑖 squared times 𝑖 and 𝑖 squared is negative one, 𝑖 cubed is negative one times 𝑖 which is negative 𝑖, and also that 𝑖 to the fourth is 𝑖 squared times 𝑖 squared, which is negative one times negative one, which is one.

Using these facts, you will get the same answer that one minus two 𝑖 to the fourth is negative seven plus 24𝑖.