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# 平方根與實數 (英): 平方根與實數

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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 the square root of 25.
- Well that's the same thing as the square root of 5 times 5,
- which is a clearly going to be 5.
- We're focusing on the positive square root here.
- Now let's do B.
- B I'll do in a different color, for 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 the 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 anymore.
- We don't see two numbers multiplied by itself here.
- So this is going to be times the square root of 6.
- Or we could even right 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.
- This part A can be expressed as the ratio of 2 integers.
- Namely 5/1.
- This is rational.
- 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 number is irrational.
- I'm not proving it here.
- This is the square root of 2 times the square root of 3.
- That's what the square root of 6 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.
- The square root of 4 is 2, leave the 6 in, and you would
- have gotten the 2 square roots of 6.
- Which you will get the hang of it eventually, but I want to
- do it systematically first.
- Let's do part 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.
- It's going to be the square root of this times
- 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.
- The square root of the 20 is 4 times 5.
- The square root of 4 is 2.
- You leave the 5 in the radical.
- So let's do part D.
- We have to do the 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 times 2
- times 5 times 5.
- Well we have one perfect square there, and we have
- another perfect square there.
- So if I just want to write out all the steps, this would be
- the square root of 2 times 2 times the square root of 2
- times the square root of 5 times 5.
- The square root of 2 times 2 is 2.
- The square root of 2 is just the square root of 2.
- The square root of 5 times 5, 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 2.
- And once again, this it is irrational.
- You can't express it as a fraction with an integer and a
- 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.
- Well let's do part E.
- The square root of 2000.
- I'll do it down here.
- Part E, 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 1000, 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 parentheses-- 2 times 2, times
- 2 times 2, times 2 times 2, times 5 times 5,
- times 5 times 5, right?
- We have 1, 2, 3, 4, 2's, and then 3, 5's, times 5.
- 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 4, this is a 4.
- So we're going to have a 4 repeated.
- And so this the same thing as the square root of 4 times 4
- times the square root of 5 times 5 times the
- square root of 5.
- So this right here is obviously 4.
- This right here is 5.
- And then times the square root of 5.
- So 4 times 5 is 20 square roots of 5.
- And once again, this is irrational.
- Well, let's do F.
- 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 1/2.
- Which is clearly rational.
- It can be expressed as a fraction.
- So that's clearly rational.
- Part G is the square root of 9/4.
- Same logic.
- This is equal to the square root of 9 over the square root
- of 4, which is equal to 3/2.
- Let's do part H.
- The square root of 0.16.
- Now you could do this in your head if you immediately
- recognize that, gee, if I multiply 0.4 times
- 0.4, I'll get this.
- But 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 16/100, right?
- That's what 0.16 is.
- So this is equal to the square root of 16 over the square
- root of 100, which is equal to 4/10, which is equal to 0.4.
- Let's do a couple more like that.
- OK.
- Part I was the square root of 0.1, which is equal to the
- square root of 1/10, 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-- 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 like you leaving that radical
- in the denominator.
- But I can already tell you that this is irrational.
- You'll just keep getting numbers.
- You can try it on your calculator, and
- it will 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 10/10.
- 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 J.
- We have the square root of 0.01.
- This is the same thing as the square root of 1/100.
- Which is equal to the square root of 1 over the square root
- of 100, which is equal to 1/10, 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.

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