# Square roots and real numbers | Pre-Algebra | Khan Academy

I have many root phrases here, or Square root phrases And what I’m going to do is I’m going to show them all and I simplify it We will talk about whether these numbers are relative Or not relative So let’s start with A A = the square root of 25 Well, this is the same as the square root of 5 x 5 Clearly, the result is 5 We focused on the positive square root here Let us now solve B I’m going to solve B in a different color, and I mean the primary root When I say positive square root B, we have the square root of 24 So what we have to do here, is that we want to get The main factors for this number So 24, let’s take a factor It is 2 x 12 12 = 2 x 6 And 6 is 2 x 3 So the square root of 24 is The square root of 2 x 2 x 2 x 3 This is equal to 24 Well, as you can see here, we have one whole square So we can rewrite it This is equivalent to the square root of 2 x 2 x The square root of 2 x 3 Obviously, these 2 This is the square root of 4 The square root of 4 = 2 It cannot be further simplified We do not see that we have two numbers that can be multiplied by themselves here This will be multiplied by the square root of 6 Or we can write it as a square root of 2x The square root of 3 As I said, I now want to determine if the numbers are Relative or not This is relative Part A can be represented as a ratio of two integers It is 5/1 So this is relative As for this, it is not relative Not relative I will not prove this now But anything that results from numbers is relative And the square of any prime number is non-relative I will not prove this now This square root of 2 x The square root of 3 This is the truth of the square root of 6 This makes it non-relative I cannot write it in broken form It cannot be expressed as an integer Another integer as I did there I will not prove this either But I earn you some practice And a quick way to do this You might say, the 4 can be divided by this 4 full box And let me take out the 4 So I have 4 x 6 The square root of 4 is 2, and we leave the 6 inside the root, so we get 2 square roots of 6 This is what you will notice, but I want to To do it in a systematic way first Let’s go to Part C. The square root of 20 And again, 20 is 2 x 10, and 10 is 2 x 5 So this is equal to the square root of 2 x 2 X 5, right? Now, the square root of 2 x 2, it’s very clear Equal to 2 That is, it will equal the square root of this x The square root of this 2 x Square Root of 5 Once again, you can do this mentally With a little practice The square root of 20 is 4 x 5 The square root of 4 is 2 And leave the 5 inside the root Now let’s solve Part D We have to find the square root of 200 We follow the same method Let’s take the basic factors of number It is equal to 2 x 100, and 100 is 2 x 50, which is the turn 2 x 25, 25 is 5 x 5 So these are the factors, and we can write them down Let me turn right a little This equals the square root of 2 x 2 x 2 X 5 x 5 Well, we have an entire square, and we have Full box here too If we want to write all the steps, this will be The square root of 2 x 2 The square root of 2 X square root of 5 x 5 The square root of 2 x 2 = 2 The square root of 2 is the square root of 2 The square root of 5 x 5, the square root of 25 It will be 5 So we can rearrange them 2 x 5 = 10 10 square roots of 2 Again, the number is not relative We cannot express it as a fraction containing integers Numerator and denominator And if you want to try to express this number It will last forever, but it will not repeat Okay, so let’s go to Part E The square root of 2000 And I’m going to do it here below Part E, Square Root of 2000 We follow the same method we did recently We will analyze the number to its prime factors It is 2 x 1000, 1000 = 2 x 500, = 500 2 x 250, 250 = 2 x 125, 125 = 5 x 25 And 25 = 5 x 5 So we finished So this is equal to the square root of 2 x 2- I’ll put it in brackets– (2 x 2) (2 x 2) (2 x 2) (5 x 5) (5 x 5), right? We have 1, 2, 3, 4 parentheses from 2 x 2, and three from 5 x 5 Now how much is this? Well, one thing you’ll notice is, you can write the output As follows, this is 4 So we have 4 iterations This equals the square root of 4×4 X square root of 5 x 5 x The square root of 5 Obviously, these are 4 And these 5 Then x squared root of 5 So 4 x 5 = 20 square root of 5 This is not relative again Not relative Well, let’s solve F The square root of 1/4, where we can consider it the same The square root of 1 / the square root of 4 And equal to 1/2 As is clear, it is a relative number It can be written as a fraction So it is relative Part G which is the square root of 9/4 The square root of 9/4 The same logic This is equal to the square root of 9 / square root For 4, which equals 3/2 Let’s solve the part H The square root of 0.16 You can do this mentally if you have already Understand the idea, if you hit 0.4 x 0.4, I will get this But I’m going to show you a more organized way of doing this, then This was not clear to you So this is equal The square root of 16/100, right? This is the 0.16 This is equal to the square root of 16 / The square root of 100, equal to 4/10, equals 0.4 Let’s solve more of these examples OK Part I, which is the square root of 0.1, is equivalent The square root of 1/10, equal 1 / square root of 10, i.e. 1 / – Now, the square root of 10– 10 is 2×5 So this did not help us much The square root of 10 remains the same Some math teachers do not like to leave a radical phrase In the place But I can tell you that this is not relative Not relative You will still get numbers You can try this using a calculator, and Will not be repeated The calculator will give you an approximation In order to give you the exact value, it must be You have an unlimited number But if you want to make it relative I will show you If you want to get rid of the root in the place So you can multiply by the square root of 10 / The square root of 10, right? This is 1 So you’ll get the square root of 10/10 These two phrases are equal, and both Not relative So we take the non-relative number, divide it by 10, and we will get On a non-relative number as well Let’s solve J J We have the square root of 0.01 This equals the square root of 1/100 The square root of 1 is also equal to 1 / square root For 100, equals 1/10, or 0.1 And again it’s a relative number It can be written as a fraction This number at the top is considered relative Because it can be written as a broken image