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Introduction to Imaginary Numbers - Problem 1

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Teacher/Instructor Carl Horowitz
Carl Horowitz

University of Michigan
Runs his own tutoring company

Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!

Simplifying a square root using imaginary numbers i. So let’s look at this problem square root of negative 32. So what we really need to do with this is divide it up to its real component, the numbers we know and its imaginary component numbers we don’t name it square roots and negatives.

So we can rewrite this then as the square root of 32 which we know how to deal with, and the square root of -1. Using rules of square roots we know that 16 a perfect square that goes in to 32. So this really we can divide up to into the square root of 16 square root 2 square root -1.

Squarer root 16 is 4 square root 2 we can’t do anything with. And we know that the square root of negative 1 is actually i so we can change this to the number i. So what we really have then is the square root of negative 32 is the same thing as for root 2 i.

Now one thing to be careful of is when we are dealing with square roots it sometimes can be hard to figure out if the 'i' is actually In the square root or not. So if I write square root of 2i not very clearly you don’t know whether that l is in the square root or outside the square root.

So in general what we do is actually put the i before the square root just to make sure that confusion doesn’t occur. So what i would actually write for this answer then is 4i root 2. This answer in theory is perfectly fine but just it could be easily misread not knowing what’s going on, so just put your i outside of the square root to make sure you know what’s going on.

So breaking up a negative square root using real and imaginary components to get the answer.

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