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Video: Multiplying Imaginary Numbers

Tim Burnham

Simplify (2𝑖)²(−2𝑖)³.

02:01

Video Transcript

Simplify two 𝑖 all squared times negative two 𝑖 all cubed.

And in this particular question, 𝑖 isn’t just any old variable; it represents the imaginary part of a complex number. So in other words, we’ve got two imaginary numbers multiplied together.

Now remember, with imaginary numbers we’ve defined, 𝑖 squared to be equal to negative one. In other words, 𝑖 is the square root of negative one.

Okay, so let’s expand out our calculation. And two 𝑖 all squared times negative two 𝑖 all cubed means two 𝑖 times two 𝑖 times negative two 𝑖 times negative two 𝑖 times negative two 𝑖, so let’s multiply these pairs of parentheses out. Two by two, well two 𝑖 times two 𝑖 equals four 𝑖 squared.

Oh remember, we said 𝑖 squared is equal to negative one, so we can replace that in this part of the calculation here. This means four times negative one or negative four. Okay, let’s take the next two parentheses along. We’re gonna do negative two 𝑖 times negative two 𝑖. Well negative two times negative two is positive four, and 𝑖 times 𝑖 is 𝑖 squared.

So that leaves us with negative four times four 𝑖 squared times negative two 𝑖. And again, we said 𝑖 squared is equal to negative one, so we can replace the 𝑖 squared with negative one. So that middle term is four times negative one, which is negative four.

Well now let’s multiply these two terms together: negative four times negative four is positive 16. So we’ve got 16 times negative two 𝑖, which just means 16 times negative two times 𝑖, and 16 times negative two is negative 32.

So we’ve got negative 32𝑖 Now we can’t actually simplify that down any further, so that’s our answer: negative 32𝑖.