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Simplify two π all squared times negative two π all cubed.
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And in this particular question, π isnβt just any old variable; it represents the imaginary part of a complex number.
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So in other words, weβve got two imaginary numbers multiplied together.
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Now remember, with imaginary numbers weβve defined, π squared to be equal to negative one.
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In other words, π is the square root of negative one.
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Okay, so letβs expand out our calculation.
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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 letβs multiply these pairs of parentheses out.
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Two by two, well two π times two π equals four π squared.
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Oh remember, we said π squared is equal to negative one, so we can replace that in this part of the calculation here.
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This means four times negative one or negative four.
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Okay, letβs take the next two parentheses along.
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Weβre gonna do negative two π times negative two π.
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Well negative two times negative two is positive four, and π times π is π squared.
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So that leaves us with negative four times four π squared times negative two π.
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And again, we said π squared is equal to negative one, so we can replace the π squared with negative one.
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So that middle term is four times negative one, which is negative four.
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Well now letβs multiply these two terms together: negative four times negative four is positive 16.
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So weβve got 16 times negative two π, which just means 16 times negative two times π, and 16 times negative two is negative 32.
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So weβve got negative 32π.
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Now we canβt actually simplify that down any further, so thatβs our answer: negative 32π.