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Multiplying Radicals of the Same Root - Concept

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!

When multiplying radical expressions of the same power, be careful to multiply together only the terms inside the roots and only the terms outside the roots; keep them separate. After multiplying the terms together, we rewrite the root separating perfect squares if possible. The rules of distributing and multiplying binomials (FOIL) apply to radicals as well.

Now we're going to talk about multiplying radicals, so we know how to simplify radicals okay so behind me I have the square root of 50. We know how to simplify this by saying okay what perfect square goes into 50? 25 so we can break this down into 25 times 2 which we can then split up again into square root of 25 times the square root of 2. Square root of 25 is simplifies the 5 so we're ending up with 5 root 2. Okay, so we're able to split it up to get a simplified version okay? If we're multiplying two things like over here I have square root of 2 times the square root of 8, as long as our square roots are the same so these are both squares so they're both to [IB] 2 right here we could go the other way as well so with this 50 we broke it down into the product of two things, we could also go the other way and put it back in together to be one square root okay? So what we can do here is combine these two together this turns into the square root of 16 which we know is 4 okay?
The other way you could do this and I wouldn't recommend it because it's little more work is to simplify these two individually okay? We can't do anything with this squared of 2 that's stuck, but the square root of 8 turns into the oops let's write where I know you can see it is equal to the squared of 4 to the square root of 2 so what we'll really end up with is the square root 2 from before times 2 times another square root of 2 okay? Square root of 2 times the square root of 2 turns into 2 so we end up with 2 times 2 which is 4 okay? So we could simplify these up first but there's really no point okay? We can combine these two as is same square root multiply them together in order to simplify.

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