Hi guys! I'm Nancy. And I'm going to show you how to simplify radicals and radical expressions. How to get them into the simplest form possible or 'simplified radical form'. We're going to focus on, specifically, square roots in this video. First I'll show you some simple ones, and then I'll show you some more complicated ones. Alright. So these are all the kinds of radical problems I'm going to show you how to simplify. We're going to start with simpler numbers, smaller numbers, under the root and I'll show you how to break them down and simplify them. Then I'll show you kinds that have a large number under the root. And how to break that down. Then I'll show you products. So how to simplify when you have a number times a square root. Then I'll show you a quotient, so a number divided by a radical. I'll show you how to get rid of the square root in the denominator. Since that's considered 'simpler', not to have it in the bottom. And then finally I'll show you how to simplify a more complicated example. Where in order to get rid of the square root in the bottom you have to simplify using something called a 'conjugate'. So, if you're working on a specific kind of problem and you see the one you're working on, you can skip ahead. Just look in the description to see what time to skip to. So let's start with one of the simpler, smaller number examples. OK. Let's start with the simplest type. Which is if the number underneath the root is a perfect square. You're going to need to know perfect square numbers. So let me show you the first 13 perfect squares. So over here I listed some of the perfect squares. What are perfect squares? It just means that each of these numbers has a number that when multiplied by itself, gives you back this number. So for 1, that's 1 x 1 = 1. For 4, it's 2 x 2. For 9, it's 3 x 3 = 9. And so on. So if the number under your square root is just a perfect square then you can simplify it immediately. You look for the number that when multiplied by itself gives you 16. And that's 4. So the answer is just 4. Similarly, if you had square root of 81. Check to see if it's on your perfect square list. And since 9 x 9 gives you 81, the answer is just 9. Alright. Say you have the square root of 32. What do you do when the number underneath the root is not already a perfect square? It's not a number on this list. It's OK. What you have to do is find a perfect square number that will divide in evenly into 32. You take 32 and you try dividing by these perfect squares until you find one that goes in evenly. Start with 4. Try 32 divided by 4. Don't start with 1, because dividing by 1 would just give you 32. So you would still have root 32. You'd be back where you started. So start with 4. 32 / 4 goes in evenly. That gives you 8. 4 x 8 is 32. So what you do, is you take those two numbers: 4 and 8 and you write each of them under their own square root symbol. So we have: sqrt(4) x sqrt(8). Now, keep simplifying as much as you can. Square root of 4, simplifies to just 2, because 4 is a perfect square. That's why you chose it. So it's just 2 in place of the root and then you still have the square root of 8. Now this is important... Always ask yourself, 'Can I simplify this root any more?' Square root of 8 does simplify more because the number underneath, 8, can be broken into 4 x 2. If you divide 8 by 4, it goes in evenly. So what you need to do is, just like before rewrite the square root of 8 part instead as: sqrt(4) x sqrt(2) Keep simplifying. Keep seeing if you can simplify more. In the next line we have 2. Square root of 4, just becomes 2. The constant 2. And then we have a sqrt(2). Can you simplify sqrt(2) even more? No, because it's not on the perfect square list. So it stays in your answer. All that you can do, is combine these two constants into one. So you have: 4 x sqrt(2) So your final simplest answer, your simplified radical form is: 4 x sqrt(2) Alright. What if you have an even larger number under the square root? A big ugly number like 125. You might be thinking, 'Whoa, I don't know how to solve that.' Actually, you do. It's the same idea as before.. you want to find a perfect square number that will go in evenly into 125. So you take 125 and try dividing them. Start with 4. 125 divided by 4 does not go in evenly. If you punch it into your calculator, you'll get a decimal. And that's how you'll know. Or you could do it by hand, off to the side. So try 9 next. 125 divided by 9 does not go in evenly. 125 divided by 16. No. 125 divided by 25.. does work. Because 25 x 5 is 125. So what you do is you take those two numbers: 25 and 5 and you write each of them under their own square root symbol. So you have: sqrt(5) x sqrt(25) Now keep simplifying in the next line. Square root of 5 can't be simplified any more because 5 is not a perfect square so the root 5 stays in your answer. Square root of 25 does simplify. It reduces to just 5. That's why you picked it, because it's a perfect square. So instead of sqrt(25), you just write 5. No root. Now, it's considered simpler to write the constant number in front of the root. People wouldn't leave it this way. So just move that number out front. So you have 5 x sqrt(5). So your final simplest answer, the simplest form simplified radical form is 5 x sqrt(5). OK. What if you have a product? A number times a square root. It's still the same idea of breaking it up into a perfect square and another number. This constant out front will stay there. The 5 is already simplified. So it stays. sqrt(54) is the one you want to break up. So see if there's a perfect square number that will go in evenly. That will divide evenly into 54. 54 divided by 4 doesn't go in evenly. 54 divided by 9 does. Because 9 x 6 is 54. So you take those two numbers, 9 and 6 and in place of the 54, you're going to write sqrt(9) x sqrt(6) Keep simplifying, just like before. And you have 5.. sqrt(9) reduces to 3, because it was a perfect square in your list. sqrt(6) does not simplify any more. There is no perfect square that goes in evenly into 6. Not even 4. So you leave it. Keep simplifying if you can. In this case you just want to combine the two constants. 5 x 3 is just 15. And the sqrt(6) is still there. So you answer is: 15 x sqrt(6). That's the simplest. Alright, let's try one more quick example that's a product. Say you have: 4 x sqrt(31) So just like before You want to keep that constant there, keep the 4. But you should try to simplify sqrt(31). Take the number underneath: 31 Try dividing by the perfect square numbers. 31 divided by 4. 31 divided by 9. None of them go in evenly. In fact, 31 itself is a prime number. So it has no factors other than 31 and 1. All you need to do it rewrite sqrt(31) in this case. So it's OK if sometimes a larger number can't broken down anymore. Sometimes the problem you're given can't be simplified any further. So the answer is, once again: 4 x sqrt(31) OK. What if you have a quotient? You have a number divided by a square root and you're supposed to simplify it. Well this looks pretty simple, but actually.. it's considered 'simpler' If you get the square root out of the denominator. You're not supposed to have a root in the bottom. So there's a trick you have to use. What you do is you take that number: sqrt(2) and you multiply this expression by that same exact sqrt(2) over itself. So you take what's in the bottom: sqrt(2) and you multiply by sqrt(2) over sqrt(2). The reason you can do this is that this value is only 1. It's just 1, so it doesn't change the overall value of the expression. But it will help you simplify and it will get that root out of the bottom. You'll see. All you need to do now is keep simplifying. So in the next line multiply straight across, this is two fractions multiplied together. So you multiply top straight across and bottom straight across. So on the top you have: 4 x sqrt(2) and on the bottom you have: sqrt(2) x sqrt(2) Keep simplifying. Next line. 4 x sqrt(2). That can't be simplified any more. On the bottom: sqrt(2) x sqrt(2) just simplifies to 2 only. You can think of it as those root symbols canceling. Or you can think of it as sqrt(2) squared is just 2. Either way, this simplifies to just 2 in the bottom. Ask yourself now, 'Is there anything else I can do to simplify this?' Actually yeah. You can do 4 divided 2. You can simplify the constants out front. So you have: 2 x sqrt(2) And that's your final, simplest form. Alright. One more type that's a little more complicated. What if you're given something like this.. that has a number plus or minus a root in the denominator. In this fraction in the bottom. You have to get rid of that root, that square root in the bottom for it to be considered simplest form. How do you do that? There's a trick you have to use and it's multiplying by the conjugate. Conjugate. What's the conjugate? It just means the bottom, but with the sign flipped. What I mean by that is.. you're going to multiply by this expression 5 - sqrt(3), but with the sign flipped. So it's actually: 5 + sqrt(3) (That's very important that you change the sign.) and over itself. The same 5 + sqrt(3) in the bottom. This looks really weird, but it will help you simplify. And it is the only way. It is the trick to get this into simplest form. The reason you can do that, is because this is just equal to 1 here. So it doesn't change the overall value of this radical expression. Now you're going to simplify. And unfortunately simplifying something like this means you first have to distribute and multiply things out. It's very helpful to put those parenthesis there so that you get all the terms you need to get. So let's multiply them out. You're going to multiply straight across on top. And you'll need to foil, distribute, all of these terms. So you get: 4 x 5 (which is 20) plus 4 x sqrt(3), which is 4 x sqrt(3) plus sqrt(3) x 5, which is 5 x sqrt(3) plus sqrt(3) x sqrt(3), which is just 3. Those root symbols basically cancel. sqrt(3) x sqrt(3) is just 3. So we have plus 3. Now for the bottom. Same thing. Foiling. 5 x 5, gives us 25. 5 x sqrt(3) is 5 x sqrt(3). So plus 5 x sqrt(3). -sqrt(3) x 5. That negative's important. Is -5 x sqrt(3). And then -sqrt(3) x sqrt(3) would just give you -3. The sqrt(3) x sqrt(3) reduces to just 3. And you do have a negative sign. Alright, so you foiled everything out. Multiplied everything out. Now just keep simplifying and hopefully things will cancel. That's the whole point of this. On top, combine all the like terms you can. The constants need to come together. Just the numbers 20 and 3, you combine into 23. 4 x sqrt(3) and 5 x sqrt(3) are like terms because sqrt(3) and sqrt(3) are attached to both so you can combine them and you just get 9 x sqrt(3) since it's adding. 9 x sqrt(3). Now the bottom. Combine like terms. Cancel when you can. 25 and -3 is 25 - 3, so you have 22. +(5 x sqrt(3)) and -(5 x sqrt(3)) just cancel. So you have nothing from those terms. Alright, so keep simplifying if you can. You have 23 + 9 x sqrt(3) all over 22. You should check to see if there's a number that goes into 23, 9 and 22. There's not. So in this case, this is your simplest answer. Why is it the simplest answer? Why is it simpler than the original? Only because there's no square root in the bottom. So the answer, simplest radical form is 23 + 9 x sqrt(3) over 22. So I hope this video helped you understand how to simplify square root problems. Simpler's always better... right? It's OK! You don't have to like math... but you can like my video! 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