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## Algebra basics

### Course: Algebra basics>Unit 1

Lesson 4: Square roots

# Simplifying square roots of fractions

We can simplify the √(1/200) by finding perfect squares that are factors of 200. We could either look directly for the largest perfect square factor or break 200 into smaller factors and find repeated factors. If we have √2 in the denominator, we can multiply by √2/√2 to make an equivalent fraction.

## Want to join the conversation?

• At Sal says that sqrt(2) * sqrt(2) is 2. How does that work?
• We can multiply square roots.
sqrt(2) * sqrt(2) = sqrt(4)
What is the sqrt(4)? It = 2.
• Just, completely lost. I've been following the fundamentals okay up until now, but this lesson makes me think Ive missed out a whole semester.
• How on earth is ONE 200th equal to a 20th of the square root of 2?
The square root of 200 is equal to a 20th of the square root of 2. Not ONE 200th.
Please explain... I, and my calculator are both confused.
• I'm not sure what you mean.
1/200 ≠ √2/20
I know this immediately because the LHS is rational and the RHS is irrational so they cannot be equivalent. What makes you think that it was implied that they were equivalent?
• I might be a little dumb but this just feels like overcomplicating things xd
• I am soo lost
• its fine just maybe do the topic test or smth
(1 vote)
• How does the square root of 2 times the square root of 2 equal 2?
• I wish he explained why he puts his symbols and numbers where he puts them. Why do you put the 10 on the outside of the radical and the 2 inside?
(1 vote)
• You can only bring items outside the radical if you can do the square root.
1) Sal starts with sqrt(200)
2) He splits it into sqrt(2)*sqrt(100)
-- The 2 is a prime number, and not a perfect square. So, it must stay inside the radical.
-- The 100 is a perfect square. 100 = 10^2. So you can do the sqrt(100) = 10. The 10 is now outside the square root.
-- Write the numbers in the correct order: numbers outside go first, then the remaining radical. This gives you: 10 sqrt(2)

Hope this helps.
• I understand square roots a lot better than most people in my class, but I'm stumped on how to simplify square roots this video made me a little more confused, can someone help break it down for me?
• Just prime factorize and find any perfect squares in the number. Then find the root of those perfect square and put that outside of the square root! I recommend making sure you are familiar with these terms as sal uses it alot in his later videos.
• Can someone explain why the two in purple is separated from the other twos?
(1 vote)
• 200 factors into 2*2*2*5*5
When simplifying a square root, you need numbers that are squared to bring a value outside the radical. Notice, Sal pairs up the factors into (2*2)(5*5)(2). He can take the square root of (2*2), it becomes 2. He can take the square root of (5*5), it becomes 5. He can't take the square root of 2 (this is the purple 2) because there is only one 2. So, it stays in the radical.
Hope this helps.
• i dont get it at all

## Video transcript

- [Voiceover] So we have here the square root, the principal root, of one two-hundredth. And what I want to do is simplify this. When I say "simplify it" I really mean, if there's any perfect squares here that I can factor out to take it out from under the radical. And so I encourage you to pause the video and see if you can do that. Alright so there's a couple of ways that you could approach this. One way is to say, Well this is going to be the same thing as the square root of one over the square root of 200. The square root of one is just one over the square root of 200. And there's a couple of ways to try to simplify the square root of 200. I'll do it a couple of ways here. Square root of 200. You could realize that, OK, look 100 is a perfect square. And it goes into 200. So this is the same thing as two times 100. And so the square root of 200 is the square root of two times 100, which is the same thing as the square root of two times the square root of 100. And we know that the square root of 100 is 10. So it's the square root of two times 10 or we could write this as 10 square roots of two. That's one way to approach it. But if it didn't jump out at you immediately that you have this large perfect square that is a factor of 200, you could just start with small numbers. You could say, alright, let me do this alternate method in a different color. You could say, ah it's the same color that I've been doing before. (laughing) You could say that the square root of 200, say Well it's divisible by two. So it's two times 100. And if 100 didn't jump out at you as a perfect square, you could say, Well that's just going to be two times 50. Well I can still divide two into that. That's two times 25. Let's see, and 25, if that doesn't jump out at you as a perfect square, you could say that that's not divisible by two, not divisible by three, four, but it is divisible by five. That is five times five. And to identify the perfect squares you would say, Alright, are there any factors where I have at least two of them? Well I have two times two here. And I also have five times five here. So I can rewrite the square root of 200 as being equal to the square root of two times two. Let me just write it all out. Actually I think I'm going to run out of space. So the square root, give myself more space under the radical, square root of two times two times five times five times two. And I wrote it in this order so you can see the perfect squares here. Well this is going to be the same thing as the square root of two times two. This second method is a little bit more monotonous, but hopefully you see that it works, (laughing) I guess is one way to think about it. And they really, they boil down to the same method. We're still going to get to the same answer. So square root of two times two times the square root times the square root of five times five, times the square root of two. Well the square root of two times two is just going to be, this is just two. Square root of five times five, well that's just going to be five. So you have two times five times the square root of two, which is 10 times the square root of two. So this right over here, square root of 200, we can rewrite as 10 square roots of two. So this is going to be equal to one over 10 square roots of two. Now some people don't like having a radical in the denominator and if you wanted to get rid of that, you could multiply both the numerator and the denominator by the square root of two. 'Cause notice we're just multiplying by one, we're expressing one as square root of two over square root of two, and then what that does is we rewrite this as the square root of two over 10 times the square root of two times the square root of two. Well the square root of two times the square root of two is just going to be two. So it's going to be 10 times two which is 20. So it could also be written like that. So hopefully you found that helpful. In fact, even this one, you could write if you want to visualize it slightly differently, you could view it as one twentieth times the square root of two. So these are all the same thing.