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I have here a bunch of radical expressions or square root expressions and what I'm going to do is go through all of them and simplify them and we'll talk about whether these are rational or irrational numbers so let's start with a a is equal to it's a square root of 25 well that's the same thing as the square root of 5 times 5 which is clearly going to be 5 order this is we're focusing on the positive square root here now let's do B the other in a different color or the principal root when we say positive square root B we have the square root of 24 so what you want to do is you want to get the prime factorization of this number right here so 24 let's do its prime factorization this is 2 times 12 12 is 2 times 6 6 is 2 times 3 so square root of 24 this is the same thing as the square root of 2 times 2 times 2 times 3 that's the same thing as 24 well we see here we have one perfect square right there so we could rewrite this this is the same thing as the square root of 2 times 2 times the square root of 2 times 3 now this is clearly 2 this is the square root of 4 the square root of 4 is 2 and then this we can't simplify any more we don't see two numbers multiplied by itself here so this is just going to be times the square root of 6 or we could even write this as the square root of 2 times the square root of 3 now I said I would talk about whether things are rational or not this is rational it can this part a can be expressed as the ratio of two of two integers namely 5 over 1 this is rational this is irrational this is irrational I'm not going to prove it in this video but anything that is the product of irrational numbers and the square root of any prime any prime number is irrational I'm not proving it here this is the square root of two times the square root of three that's what the square root of six is and that's what makes this irrational I cannot express this as any type of fraction I can't express this as some integer over some other integer like I did there and I'm not proving it here I'm just giving you a little bit of practice and a quicker way to do this you could say hey 4 goes into this 4 is a perfect square let me take a 4 out this is 4 times 6 square root of 4 is 2 leave the 6 in and you've got in the 2 square roots of 6 which would have got it well you'll get the hang of it eventually but I want to do it systematically first let's do part 2 C square root of 20 once again 20 is 2 times 10 which is 2 times 5 so this is the same thing as the square root of 2 times 2 right times 5 now the square root of 2 times 2 that's clearly just going to be 2 so it's going to be 2 it's going to be the square root of this times the square root of that 2 times the square root of 5 and once again you could probably do that in your head with a little practice square root of 20 is 4 times 5 square root of 4 is 2 you leave the 5 in the radical let's do Part D Part D we have to do the square root of 200 square root of 200 same process let's take the prime factors of it so it's 2 times 100 which is 2 times 50 which is 2 times 25 which is 5 times 5 so this right here we can rewrite it let me scroll to the right a little bit this is equal to the square root of 2 times 2 2 times 2 times 2 times 5 times 5 well we have 1 perfect square there and we have another perfect square there so if we I just want to write all the steps this would be the square root of two times two times the square root of two times the square root of five times 5 square root of 2 times 2 is 2 square root of 2 is just the square root of 2 square root of 5 times 5 square that's the square root of 25 that's just going to be 5 so you can rearrange these 2 times 5 is 10 10 square roots of 12 2 and once again this is irrational you can't express it as a fraction of with an integer and a in the numerator and the denominator and if you were to actually try to express this number it will just keep going on and on and on and never repeating let's do Part II square root of 2000 I'll do it down here part-ii the square root of 2000 same exact process that we've been doing so far let's do the prime factorization that is 2 times 1,000 which is 2 times 500 which is 2 times 250 which is 2 times 125 which is 5 times 25 which is 5 times 5 and we're done so this is going to be equal to the square root of 2 times 2 I'll put it in parenthesis 2 times 2 times 2 times 2 times 2 times 2 times 5 times 5 times 5 times 5 all right we have one two three four twos and then three five times five now what is this going to be equal to well one thing you might see is hey I could write this as this is a four this is a four so we're gonna have a four repeated and so this is the same thing as the square root of four times four times the square root of five times five times the square root of five so this right here is obviously for this right here is five and then times the square root of five so 4 times 5 is 20 square roots of 5 and then once again this is a rational irrational now let's do F square root of 1/4 F the square root of 1/4 F was the square root of 1/4 so f is the square root of 1/4 which we can view this is the same thing as the square root of 1 over the square root of 4 which is equal to 1 over 2 which is clearly rational it can be expressed as a fraction so that's clearly rational part G part G is the square root of 9 fourths square root of 9 over 4 same logic this is equal to the square root of 9 over the square root of 4 which is equal to 3 over 2 let's do part H part H the square root of 0.16 now you could do this in your head if you immediately recognize that G if I multiply 0.4 times 0.4 I'll get this but I'll treat you or I'll show you a more systematic way of doing it if that wasn't obvious to you so this is the same thing as the square root of 16 over 100 right that's what point 16 is so this is equal to the square root of 16 over the square root of 100 which is equal to 4 over 10 which is equal to 0.4 let's do a couple more like that okay Part I was the square root of 0.1 which is equal to the square root of 1 over 10 which is equal to the square root of 1 over the square root of 10 which is equal to 1 over now the square root of 10 that's a 10 is just 2 times 5 so that doesn't really help us much so that's just the square root of 10 like that a lot of math teachers don't let you don't like you leaving that radical in the denominator but I can already tell you that this is irrational irrational this will just keep a lot you'll just keep getting numbers you could try it on your calculator and we'll never repeat your calculator will just give you an approximation because in order to give the exact value you'd have to have an infinite number of digits but if you wanted to rationalize this just to show you if you want to get rid of the radical in the denominator you can multiply this times the square root of 10 over the square root of 10 right this is just 1 so you get the square root of 10 over 10 these are equivalent statements but both of them are irrational you take an irrational number divide it by 10 you still have an irrational number let's do a J J we have the square root of 0.01 this is the same thing as the square root of 1 over 100 which is equal to the square root of 1 over the square root of 100 which is equal to 1 over 10 or 0.1 clearly once again this is rational it's being written as a fraction this one up here was also rational it can be written expressed as a fraction