# Question Video: Multiplying Imaginary Numbers Mathematics • 12th Grade

Simplify (2π)Β²(β2π)Β³.

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### Video Transcript

Simplify two π all squared times negative two π all cubed.

And in this particular question, π isnβt just any old variable; it represents the imaginary part of a complex number. So in other words, weβve got two imaginary numbers multiplied together.

Now remember, with imaginary numbers weβve defined, π squared to be equal to negative one. In other words, π is the square root of negative one.

Okay, so letβs expand out our calculation. And two π all squared times negative two π all cubed means two π times two π times negative two π times negative two π times negative two π, so letβs multiply these pairs of parentheses out. Two by two, well two π times two π equals four π squared.

Oh remember, we said π 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, letβs take the next two parentheses along. Weβre gonna do negative two π times negative two π. Well negative two times negative two is positive four, and π times π is π squared.

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

Well now letβs multiply these two terms together: negative four times negative four is positive 16. So weβve got 16 times negative two π, which just means 16 times negative two times π, and 16 times negative two is negative 32.

So weβve got negative 32π. Now we canβt actually simplify that down any further, so thatβs our answer: negative 32π.