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This is going to be an introduction to square roots and radicals.
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I'll be explaining what those are in just a minute.
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But let's begin with a review of what it means to square a number.
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So for instance we know 0 square just means 0 times 0,
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which is 0. Let's skip to something like 3 square.
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That would mean 3 times 3 or 9.
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Let's say we have a number in parenthesis squared like negative 2.
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That would be negative 2 times negative 2, which is positive 4.
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So remember when you're squaring a negative number, it's going to be positive.
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Let's take a fraction like 5, 6 squared.
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That would mean 5, 6 times 5, 6, which is 25, 36.
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Let's take a decimal like 0.4 squared.
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This would mean 0.4 times 0.4.
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And so we would do 4 times 4, which is 16.
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And then we need to move in a decimal place for each of these.
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So this is going to be 0.16.
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So for instance this is just a little review of what it would mean to square a number.
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And of course there's infinitely many numbers you can square.
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I'd recommend knowing up to let's say maybe 15 squared of all the whole numbers.
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So using your times tables, once you get up to 10 squared,
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you should know that's 11 squared, 11 times 11 is 12 squared is 144.
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Most people know this from your times tables, but let's just do 3 more,
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which I don't think you'll find too hard.
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And you don't have to memorize these, but it might be convenient.
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13 squared is 169, and 14 squared is 196.
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Notice you just switched the last two digits from 13 squared to 14 squared.
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So if you want to memorize those, those aren't too bad.
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And 15 squared is 225, 15 times 15.
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If you don't memorize them, that's fine, but they might come in handy if you do know those.
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There's a couple more.
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What about if you have something with zeros at the end?
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Like how about 30 squared?
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That would just be 30 times 30.
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And just as a reminder, when you have something with zeros,
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just multiply the three times three.
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And then you put a zero for each one in there, so you get 900.
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So if you have something with more zeros, that's the little trick.
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So if you have 50 squared, you're just going to do the five times the five, right?
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Because it's 50 times 50, which is going to be 25, and then two extra zeros.
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So this is just a little introduction to how to square numbers.
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Now, let's take a couple of things here.
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Let's do three squared, which we see is nine.
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And then we're going to do a negative three, but in parentheses squared.
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So that's negative three times negative three.
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And notice you also get nine.
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Now, if you square a number, and you get nine, what this means is that that number that you squared is a square root of nine.
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So what we have is both three, oops, very good handwriting, both three and negative three are called square roots of nine.
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Because when I square them, I got nine.
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So a number is called a square root of a number, if when you square it, you get that number.
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So both three and negative three are square roots and negative nine.
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Now, what if I asked you?
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What are the square roots of 16?
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So it's just a question, what are the square roots of 16?
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You have to think, well, what number when I square it will give me 16?
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And actually there are 10 numbers, four and negative four are both called square roots of 16.
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Now, here's the tricky part.
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If you write a symbol that we use for square root, so the square root symbol looks like this.
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And what you do is you put the number underneath it.
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So now check this out.
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Square root of 16.
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You might think, well, what am I going to say the answer to that is, is it four or is it negative four?
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Because when I square four, I get 16 and when I get, when I square negative four, I get 16.
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So square root of 16 means following the principal square root of 16.
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Okay, all right.
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So now, well, what's that mean?
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So the key is the difference here is that I'm asking for the principal square root.
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And the principal square root means the non-negative square root.
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All right.
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So the principal square root means the non-negative one.
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So it could be zero like the square root of zero or it's going to be a positive number.
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Okay, so let's go on.
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So what's the square root of 16?
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If I use the symbol, so now I'm not just asking you, hey, what are the square roots of 16?
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I'm actually writing this symbol for the square root.
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Your answer cannot be negative.
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That's what it means.
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So we would only choose the number four.
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So when we use this symbol, that means there's really only going to be one answer.
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So what would the square root of 9 be?
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You have to think, well, what number times itself is 9?
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Well, you could use three in negative three, but again, we're using this symbol, so we want the non-negative square root,
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which is going to be positive three.
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So here's the difference.
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And this is very kind of confusing.
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So if I say find all square roots of 81, this would mean, well, 9 and negative 9.
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And finding all the square roots of 36, you could say 6 and negative 6.
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But if instead the directions I'd find the following and I'm using that square root symbol, so this means the square root of 81,
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you would just pick 9 and the square root of 36 would just be 6.
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So it's a subtle difference, but when you sort of have to be aware of, and for the most part, when you're working in algebra,
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you'll be using the symbol, the square root symbol.
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So when you take the square root of a number, you will just always choose the positive number, the principal square root, which is non-negative.
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Just keep in mind that I don't say it's always positive because if I did say what's the square root is 0, that's 0.
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And that's not a positive number, but it is non-negative, right?
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Let's say you were given this problem, evaluate square root of negative 9, you think, okay.
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What numbers can I multiply by itself? What number can I multiply by itself and get negative 9?
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But remember when we began, when you squared a number, it was positive.
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So in general, when you square a number, you're going to get positive.
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I'm going to be careful here. For all the real numbers, like all the positive and negative numbers, which is all we're working with right here, it's not negative.
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So what we would say is, square root of negative 9 is not a real number.
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And I'm not going to say that it's not an answer because as you go on in algebra, you'll find out that there is something else, which we'll call a complex number.
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So we're going to be able to evaluate this later on.
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And some course, maybe not the one you're in right now.
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But for now, the square root of negative 9 is not a real number.
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So I want you to try a couple of problems now.
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All right, put the video on pause and try these four problems.
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All right, here we go. This is the first one. Square to 25.
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What times itself is 25? 5.
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All right, next one. Negative square to 49.
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Now, notice the negative sign is out in front of the radical sign.
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This symbol, which I call the principal square root, is also called the radical sign.
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So I'm going to say that that is minus. I'm just going to write the minus sign.
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And then the square root of 49 is 7. So negative, the square root of 49 is the same as negative 7.
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Square root of negative 64. Uh-oh. It doesn't equal any number.
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So instead of writing equal, I'd say is not a real number.
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Because I've got the square root of a negative number.
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And last one, square root of 9, 16. So we're thinking, hey, uh-oh, what's that one?
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What times itself is 9, 16? We'll give you a little hint.
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If they're what we call perfect squares in the numerator and the denominator,
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like square to 9 is 3 and square to 16 is 4.
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You just take the square root of the top, which is 3 and the square root of the bottom, which is 4.
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So this gives you 3-4s.
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And so that's just a little introduction to square roots and radicals.
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So this symbol here is sometimes called a radical sign and we'll be working with some other ones that look like this with a 3 in it,
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or a little 4 in this little area, etc.