Question Video: Multiplying Imaginary Numbers Mathematics

Simplify (2๐‘–)ยฒ(โˆ’2๐‘–)ยณ.

02:00

Video Transcript

Simplify two ๐‘– all squared times negative two ๐‘– all cubed.

And in this particular question, ๐‘– isnโ€™t just any old variable; it represents the imaginary part of a complex number. So in other words, weโ€™ve got two imaginary numbers multiplied together.

Now remember, with imaginary numbers weโ€™ve defined, ๐‘– squared to be equal to negative one. In other words, ๐‘– is the square root of negative one.

Okay, so letโ€™s expand out our calculation. And two ๐‘– all squared times negative two ๐‘– all cubed means two ๐‘– times two ๐‘– times negative two ๐‘– times negative two ๐‘– times negative two ๐‘–, so letโ€™s multiply these pairs of parentheses out. Two by two, well two ๐‘– times two ๐‘– equals four ๐‘– squared.

Oh remember, we said ๐‘– 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, letโ€™s take the next two parentheses along. Weโ€™re gonna do negative two ๐‘– times negative two ๐‘–. Well negative two times negative two is positive four, and ๐‘– times ๐‘– is ๐‘– squared.

So that leaves us with negative four times four ๐‘– squared times negative two ๐‘–. And again, we said ๐‘– squared is equal to negative one, so we can replace the ๐‘– squared with negative one. So that middle term is four times negative one, which is negative four.

Well now letโ€™s multiply these two terms together: negative four times negative four is positive 16. So weโ€™ve got 16 times negative two ๐‘–, which just means 16 times negative two times ๐‘–, and 16 times negative two is negative 32.

So weโ€™ve got negative 32๐‘–. Now we canโ€™t actually simplify that down any further, so thatโ€™s our answer: negative 32๐‘–.

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