Main content

## Algebra (all content)

### Unit 11: Lesson 8

Radicals (miscellaneous videos)- Simplifying square-root expressions: no variables
- Simplifying square roots of fractions
- Simplifying rational exponent expressions: mixed exponents and radicals
- Simplifying square-root expressions: no variables (advanced)
- Intro to rationalizing the denominator
- Worked example: rationalizing the denominator
- Simplifying radical expressions (addition)
- Simplifying radical expressions (subtraction)
- Simplifying radical expressions: two variables
- Simplifying radical expressions: three variables
- Simplifying hairy expression with fractional exponents

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# Simplifying square roots of fractions

Sal rewrites √(1/200) as 1/(10*√2) and as √2/20.

## Want to join the conversation?

- At4:19Sal says that sqrt(2) * sqrt(2) is 2. How does that work?(14 votes)
- We can multiply square roots.

sqrt(2) * sqrt(2) = sqrt(4)

What is the sqrt(4)? It = 2.(21 votes)

- 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.(7 votes)- 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?(4 votes)

- 4:15How does the square root of 2 times the square root of 2 equal 2?(4 votes)
- im lost on how you get 10 if 10 times 10 is 100 not 200 thats where i got lost(3 votes)
- you forgot about the other 2 that sal factored out.(1 vote)

- Can someone explain why the two in purple is separated from the other twos?(2 votes)
- 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.(2 votes)

- 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?(2 votes)
- 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.(2 votes)

- How do you make a square root into a fraction? Like 2√7?(2 votes)
- Any number that is not a fraction can be made into a fraction by using a denominator of 1.

(2√7)/1

Hope this helps.(1 vote)

- what do you do when it looks like this...7^3/^5?(2 votes)
- This is kind of a wrong statement that you are giving , the fifth power you are raising must have a base but here it does not have any .

This statement is like 7 raised to the power 3 divided by 'something' raised to the power 5 , but what is that 'something' , thus you are giving an incomplete statement .(1 vote)

- Here's what I did and I'm not sure where I went wrong

√1/200

=√1/(10*10*2)

=(1/10)*(√1/2)

or 1/(10√12)

Where did I go wrong? Or how is this not a right answer?(1 vote)- Here's how I would do it.

Split the fraction into 2 radicals.

√(1/200) = √(1)/√(200)

Simplify both square roots.

√(1) = 1

√(200) = √(10*10*2) = 10√(2)

So, your fraction becomes: 1/[10√(2)]

Next, a simplified radical will have no radicals in the denominator. So you need to rationalize the denominator.

1/[10√(2)]

= 1/[10√(2)] * √(2)/√(2)

= √(2)/[10√(4)]

= √(2)/[10*2]

= √(2)/20

Hope this helps.(3 votes)

- What would you do if you have a fraction?(1 vote)
- I will restate the video. Example: sqrt(1/200) which is equal to sqrt(1) / sqrt(200) the sqrt of 1 is 1 so 1/sqrt(200) the sqrt(200) = sqrt(2) * sqrt(100) further simplifying sqrt(100) is 10 so sqrt(2)*10 equals the sqrt(200) so we reinsert the simplified version of sqrt(200) which will end up as 1/sqrt(2)*10 which is the answer. The video also explains sqrt in the denominator so that will help more.(3 votes)

## 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.