Welcome back, everyone. So as we've talked about square roots and cube roots, we've seen lots of perfect powers, like perfect squares or cubes. So, for example, like the square root of 9, which is 3, or the cube root of 8, which is 2. But in a lot of problems, that's not going to happen. In a lot of problems, you might see something like the square root of 20 or the cube root of 54, and you're going to have to know how to take those expressions and make them simpler. That's what I'm going to show you how to do in this video, and it turns out that one of the ways that we can simplify radicals is actually by expanding them as sort of, like, making them a little bit bigger so that we can hopefully make it smaller and simpler later on. I'm going to show you how to do this. It's very straightforward. Let's get started. So, basically, when radicals aren't perfect powers like the radical twenty, the whole thing we're going to do is we're going to simplify it by turning it into a product. We're going to try to break it up into a product of 2 things, and the whole goal is that one of the terms will be a perfect power. So here's the thing. I'm going to take the radical 20, and and I want to break it up so that it's the product of 2 things, and I want one of these things to be a perfect power, like 4 or 9 or 16 or something like that. Alright? So I have this table here. We're going to just go down this table and see if we can turn the 20 into a product where 4 or 9 is one of the terms. So can this happen? So can I do 4? Well, if you take a look at 44 times 5, that equals 20. So, basically, you just separated this thing into 2 radicals, and I can totally do that. So what's the square root of 4? We have already seen that's just 2. What's the square root of 5? Well, that's just the square root of 5, and that's not a perfect number. So, basically, what we've seen here is that we've turned this into a 2 times radical 5, and so the simplest way that we can rewrite this expression is just 2 radical 5. Now can we go any further? No. Because 5 is just a prime number, so we can't break that radical up any further. So we say this expression is fully simplified because we can't break up the radicals any further than we already have. Alright? But that's the basic idea. So as a formula, the way that you're going to see this in your textbook is if a number in a radical has factors a and b, you basically can just break it up into a and b, and then you can split them up into their own radicals, like radical of of a and then radical of b. And then you can just deal with those separately. Alright? That's the whole thing. Let's move on to the second problem now because in some problems, you're going to have variables as well. So what I like to do is I like to separate this thing into the number times the variable. Alright? So, again, when I take these radicals, can I break them up into anything in which we're going to get a perfect square out of it? Well, let's do the 18 first. So could does does 18 reduce to anything? So does 4 go into 18? Well, 4 times 4 is 16. 4 times 5 is 20, so it doesn't. What about 9? Well, actually, 18 could be written as product of radical 9 times radical 2. Right? That separates. And then what about the x squared? Well, I have the square roots of x squared. So it turns out that, actually, the square root of x squared is a perfect power or sorry. The x squared is a perfect power. And, basically, what happens is you've just undone the exponents. So this actually just turns into an x over here. Alright? Now are we done yet? Is this our full expression? Well, no, because the square root of 9 actually just turns into a 3. So our what about the radical 2? Can we break up the radical 2 any further? No. Because it's just a prime number. And so, basically, what happens is this is our simplest that we can write this expression, and what you're going to see here is that the x usually gets moved in front of the radical. So this whole thing really just becomes 3x radical 2, and that is our fully simplified expression. Now for the last one over here, we have the cube root of 54x to the 4th power. So now we no longer have square roots, we have cube roots, but the idea is the same. And, again, what I like to do here is break this up into 2 radicals. So radical 54, cube root 54, and then the cube root of x to the 4th power. And I'll just deal with those independently. Right? So let's do the 54 first. Can I break this up into a perfect cube? Perfect cubes are going to be over here, like 827 and stuff like that. So what about eights? Well, 8 doesn't go into 54. 8 times 7 is 56, so it's close. What about 27? 27 is actually, yeah, this does work. So in other words, this is just 27 ⨉ 2. So I've gotten a perfect cube out of this. Alright? Now what about the cube root of x to the 4th power? Is there a perfect cube that I can pull out of that? Well, think about it. This is just x multiplied by itself 4 times, so what I can do is I can just split this up into x cubed times just cube root of x. And the reason this is helpful is because if I have a cube root of a cube, then I just basically undo it. Right? So with this 27, the cube root of this turns turns out to be is just 3. Now the cube root of 2 doesn't simplify, but what about the cube root of x to the third power? This actually just becomes x, just like the square root of x to the second power became x. And then finally, the cube root of x over here is just left alone. Alright. So how do I make this simpler? Well, I just mash these 3 and the x together and just becomes 3x. And And one of the things you'll you'll see is that when you have these two expressions, you basically just put them back together again on the under the same radical. This just becomes the, cube roots of, this just becomes 2x over here. And this is the fully simplified expression. I can't break this up any further. Alright? So that's all there is to it, folks. Let me know if you have any questions. Let's get some practice.
Simplifying Radical Expressions: Study with Video Lessons, Practice Problems & Examples
Created using AIUnderstanding how to simplify radicals is essential in mathematics. To simplify expressions like , break them into products where one factor is a perfect power, such as and . This leads to . When dealing with fractions, apply similar principles to simplify both the numerator and denominator separately. Combining like radicals requires matching radicands and indices, ensuring efficient calculations.
Expanding Radicals
Video transcript
Simplify the radical.
Simplify the radical.
Radical Expressions with Variables
Video transcript
Everyone. Up until now, all of the radicals that we've seen in our problems have always just involved only numbers. For example, like the square root of 9 or the cube root of 8. But in some problems, you might start to see variables now. You might see something like a radical x to the third power, or you might even see numbers and variables. What we're going to see in this video is that the way that we handle radicals with variables is exactly how we've been doing it with just numbers. We can apply everything that we've seen from square roots, nth roots, and even simplifying whenever we have variables and radicals. Let's go ahead and get started. We'll do a couple of examples together. So when we have radicals without variables, like just numbers, remember that the square root of 9 was 3 because 3 squared was equal to 9. These were basically the opposites of each other. So when I had the cube root of 8, that's a number multiplied by itself 3 times that gets me 8, and that was 2. That's because 2 times itself 3 times gives me 8. The idea is the same. Imagine I had something like or if I had, you know, a radical x squared. What that's asking me to do is, what is a thing that multiplies by itself to get me x squared? And it turns out to just be x. If I just take x and multiply by itself, I get x squared. So that means that the square root of x squared is just x.
Now let's say I have, like, the cube root of x to the 6th power. What's something if I multiply it by itself 3 times that gets me x to the 6th? Is it x? Well, no. Because x cubed is only going to give me x cubed, so that doesn't work. But what about something like x squared? If I take x squared and cube that, then one of the things I can use is an old exponent rule where I can merely multiply their exponents and I get x to the 6th. So that means that the cube root of x to the 6th power is just x squared.
Let's take a look at a couple of examples so I can show you how this stuff works. So here I have the square root of x cubed. This is a little bit different from the example that we worked up here where we had the square root of x squared. Basically, I want to find something where if I multiply it by itself, I get to x cubed. What if I take x and square that? That's just x squared. Well, what about x squared squared? If you work this out, what you're going to get is you're going to get x to the 4th power, which is too big. Remember, we're trying to get x to the 3rd power, so we've gone too far. So none of these will work, and it's because this thing isn't a perfect square. How do we deal with this when we had numbers? We split them, and the hope was that one of the factors was going to be a perfect power. How do I split x cubed? I can do radical of x squared and the radical of x. This radical x squared just turns out to be x. What you're left with is you're left with 1 power of x. So this is how you take radical x cubed and simplify it.
For our last question here, we have numbers and variables. It's no different. If you have radicals with numbers and variables, you can split them out into their own separate radicals, akin to how we did with x's, and then you can simplify them separately. How do we deal with radical eights? Well, remember, we split this out into radical 4 times radical 2. Radical 4 is a perfect square here, so this just turns out to be 2 times radical 2. What does the x to the 5th turn out to be? There's an index of 2. How many times does 2 go into 5? It goes in twice. And so what we do here is we just pull out a factor of this is x squared times the square root of x. So now we just put all of this back together again because all these things multiply to 8x to the 5th. So I'm going to take the x squared and put it over here. This just becomes 2x squared, and now I have a radical 2 and then a radical x. So if you take this and multiply it out by itself, you should get back to 8x to the 5th power. So that is your final answer
Simplify the radical.
Radical Expressions with Fractions
Video transcript
Welcome back, everyone. We saw how to simplify radicals by using this rule over here, which is if I can take a term and break it up into a product, then I could basically split off each term into its own radical, and I was hoping that one of these things was a perfect square and would simplify. I'm going to show you in this video that we can do the exact same thing when it comes to radicals with fractions. We can basically split up the fraction so that one of the terms might become simpler. Let me show you how this works. So we can split up or combine radicals with fractions by using these rules. We'll use the rule that we've already used before and this new one here, which is actually very similar to this. Basically, you can break up a numerator and a denominator into their own terms. It's kind of like how we broke up this into 2 different things with radicals. We can take a fraction and break it up into 2 different radicals. The hope is that one of these things will become a perfect square. Let me show you how this works. Let's say you had something like 49 over 64. Now this fraction over here, 49 over 64, if I want to divide this first, it would be really difficult because I'm going to have to think about the factors of 49, and there's actually no common factors between 49 and 64. So it's really hard to reduce this fraction. What I do notice, however, is that 49 and 64 are both perfect squares. They're both actually perfect squares over here as we can see. So what I can do is I can just break up this radical. I can break up this radical into √49 and √64, and these are actually very easy. √49 is just 7, and √64 is just 8. Alright? Let's take a look at another example here, √32 over √2. So if I want to do the same thing over here, √32 is really difficult for me to evaluate, and so is √2. Those are not perfect squares. However, what I can do here is I actually can sort of combine them into one radical and so and say that this is a √(32 over 2), and this just becomes √16. And √16 is a perfect square. That just evaluates to 4. So here's the whole point here. I actually like to think of these equations not as a one-way street. It's not always that you go from left to right. Sometimes it's better to go from right to left. So sometimes it's better to split and simplify like we did in this, example over here, and then sometimes it's actually better to divide first, and then you simplify over here. Alright? Alright. So that's all there is to it. Let's go ahead and take a look at a couple more examples because sometimes you might have variables involved instead of just numbers. So, let's take a look at this first one. √(64x^4 over 9x^2), all of that underneath one radical. So if we try to divide this first, what's going to happen is 64 over 9 isn't going to give me a clean number. But I do notice that 64 and 9 are both perfect squares, so let's try to break them up into their own radicals. This just becomes √64x^4 divided by √9x^2. Alright? So 64 is a perfect power, and x^4 is also a perfect power. It's a perfect square. So this 64 is just 8^2, and this x^4 is just (x^2)^2. Right? So using the power rule. So, basically, what this whole thing actually just becomes, 64x^4, is it just becomes x 8x^2. That's the square root of that. What does 9x^2 become? Well, 9 is a perfect square of 3, and the x^2 is just a perfect square of x^2. So in other words, this whole thing actually just becomes 3x over here. So I've taken this whole messy radical, and I've actually seen that both of them, actually, the top and the bottom, are both perfect squares of something. And I've simplified this now to basically just a bunch of, you know, letters and numbers, like 8x^2 and 3x. Now is this fully simplified? Well, actually, not quite because we have numbers here on the top and the bottom, but we also have powers of x. So this actually just really just becomes an exponents problem. Basically, I'm going to use the quotient rule, and what this answer becomes is it just becomes 8 over 3 x. Alright? Basically, just delete one power of x on the top and the bottom, and all you're left with is 8 over 3 x. Alright? That that's all there's to it. Now let's look at the second one here, 72 divided by 9. If I try to do this and try to sort of treat them as independent, what happens is the √9, that is a perfect power, but x isn't. And 72 isn't a perfect power, and we also have this x^3 over here. So it's going to be kind of tricky to sort of separate this and deal with them separately. So let's just try to combine them all under one radical and see what happens. So I'm just going to combine this as so 72x^3 over 9x. And then, basically, what happens is we're going to do the division first before we actually do the radical. So what is 72 over 9? This just becomes 8. And then what is x^3 over x? This is basically what we just did over here with the exponent rule. This actually just becomes x^2. Basically, it's like we're just doing we're canceling out one power of x. So this is basically what we're left with, √(8x^2). Can we simplify this? Well, if you notice that 8 can be broken up into a perfect square because 8 factors into 2^3, and the x^2 can also be factored out as a perfect square as well. So we're not quite done yet. We basically just have to split this out into, √8 times √x^2, and we'll deal with those separately. √8 just becomes √4 times √2. We saw that from the previous video. And then this √x^2 actually just factors out into one power of x. This is a perfect square. So this just simplifies to 2. We can move the x to the front, and this just becomes 2x√2. Now is this fully simplified? Yes. Because we can't, we can't factor anything else out. So this is basically what this whole expression becomes. Alright. So that's it for this one, folks. Let me know if you have any questions. See you in the next one.
Adding & Subtracting Like Radicals
Video transcript
Hey, everyone. We've talked a lot about radicals so far. One of the things that we haven't seen yet is how to add and subtract expressions that have radicals. That's what we're going to go over in this video. It turns out that it's actually very similar to how we dealt with algebraic expressions, and I'm going to show you it's very straightforward. So let's check it out here.
So when we dealt with algebraic expressions, if I had something like x2+3 plus 8, the way I simplified this was I combined like terms. I could combine the 2xs and the 4xs and the 3s and the 8s, and this just became 6x plus 11. The idea is that when I have a radical expression, instead of combining like terms, I'm gonna combine like radicals. So like radicals just means that they have the same radicand. They have the same thing inside of the radicals, in other words, x and x in this problem, and they also have the same index. So I have to make sure that they're both square roots and not that one is a square root and one is a cube root or something like that. As long as we have the same radicand and the same index, we can just add them. And how do we add them? We add them exactly how we added 2x and 4x. So 2 times the square root of x and 4 times the square root of x. It's kind of like I'm adding apples and apples. Right? So this just becomes 6 times the square root of x, so that's 6√x, and then the 3 and 8 just become 11 how they like, just like they always have. So this is how you simplify these kinds of expressions. You can only combine things that are like each other. Alright? That's all there is to it. So let's go ahead and take a look at a couple more examples here.
Alright. So if we have, let's say, 3 √7 times 2 plus 2 square root of 7 minus the cube root of 7, how do I simplify and add this expression? Well, remember, I can only combine the like radicals. Like radicals have like radicands and the same index. They have the same thing under the radical and the same index. Here, what we have is we'll notice that all of the radicands are 7s. I have sevens in all of the symbols over here. But are all the indexes the same? Well, no. Because here I have an index of 3, whereas here I have square roots. And remember, those are indexes of 2. So these two things have the same radicand and the same index, but this one has the same radicand but a different index. So it's not a like term. Okay? So just be very careful when you're doing that. So basically, what happens is I can combine the 2 things in yellow. So the 3 and the 2 just combine down to 5 radical 7 or square root of 7, and then I have over here minus the cube root of 7. So this is like an apple, and the cube root of 7 is like a banana. I can't add those things because they're not the same. And so this is how I simplify my expression, and that's the answer. Alright? So pretty straightforward. Now what I want to warn you against, actually, is something that I see a lot of students get this you know, make this mistake. Basically, whenever you see radicals that are, you know, separate from each other and you're adding them, you can't combine them into one radical. So, for example, I can't take the square root of 7 and the square root of 7. That's not equal to the square root of 14. This is a mistake I see a lot of students make. Just be very careful that you don't do this. Otherwise, you're gonna get the wrong answer. Radical 7+7 does not equal radical 14. You can't just merge that stuff into the same radical. Alright? Just be very careful.
Alright. So let's look at another example here. Here, we have 9 times the cube root of x, and then we have a square root of x. So, here, we have to combine the like terms. So, if you notice here, I've got the same radicands. I've got xs everywhere, but I have different indexes. Here, what I have is I have a cube root of x and a cube root of x, and then here I have a square root of x, so those are different. They're not like radicals. Then and I just have a constant over here. So what can I combine? I can combine the 9 and the 4, 9 minus 4. Remember, just keep the sign over here. This becomes 4 times the cube root of x. Then I have minus the square root of x, then I have plus 3. So that's how to simplify this expression. Alright? So that's all there is to it, folks. Let me know if you have any questions, and thanks for watching.
True or False:
and are equal.
True
False
Cannot be determined
Adding & Subtracting Unlike Radicals by Simplifying
Video transcript
Everyone, earlier in our videos, we saw how to add and subtract like radicals. For example, if I take \(3\sqrt{5}\) and \(4\sqrt{5}\), I have the same number under the same index, so I can combine them. This \(3\) and \(4\) just becomes \(7\). You just add the little numbers in front, and I have \(7\sqrt{5}\), and that is the answer. What I'm going to show you in this video is that sometimes you'll have radicals and square roots that aren't similar, like \(\sqrt{5}\) and \(\sqrt{20}\), and you're going to have to add them. I'm actually going to show you how we do that in this video, and we're going to use some old ideas that we've already seen before. Let's go ahead and get started. The idea is that when you're adding/subtracting radicals that aren't alike, you're going to have to simplify them first. So, simplify before you end up combining them like radicals. Here's the idea. Can I simplify \(\sqrt{5}\)? In other words, can I pull out a perfect power? Well, no, because the only factors of \(5\) are \(1\) and \(5\). Can we take the \(20\) and break that down into a perfect power? Well, actually, yes, we can because we saw how to do this. I'm going to take the \(20\), and I'm going to break this down into a product. And the hope is that one of the radicals ends up being a perfect power. So, we looked at \(4\), \(9\), and \(16\) all the way up to \(20\), and we saw that \(16\) and \(9\) didn't work because they don't go into \(20\), but \(4\) does. So, in other words, I can take this radical \(20\) and break it up into the product of \(\sqrt{4}\) and \(\sqrt{5}\). So why is this helpful? Well, because this just becomes \(\sqrt{5} + 2\), and this whole thing just becomes \(2\sqrt{5}\). So in other words, I've ended up or started with two square roots that were unlike each other. But by simplifying down, now I've turned it into a problem where I have the same radicands in the same index. In other words, I started out with unlike radicals. And if you simplify it, you'll end up with an expression that is like radicals. And this basically just turns into the problem on the left. So, how do I add these things? Well, \(\sqrt{5} + 2\sqrt{5}\). You just add the numbers in front. This is just \(3\sqrt{5}\), and that's how to solve these kinds of problems. Alright? So, break them down before you start combining them. That's really all there is to it. Let's go ahead and take a look at a couple more examples here. Here, we're going to do \(5\sqrt{2}\) and \(\sqrt{18}\). Alright? So what happens is, can I simplify \(\sqrt{2}\)? Well, no. Well, the whole idea is that I want these two square roots to eventually be the same because then I can subtract them. So I can't simplify \(\sqrt{2}\), but can I simplify the \(\sqrt{18}\)? And we saw how we can do that by pulling out a perfect square. In this case, \(4\) isn't going to work, but \(9\) will. So, I'm going to break this up into a product of two radicals. \(9\) goes into \(18\), and I get \(2\) as what's left over. So again, what we end up with here is we end up with \(5\sqrt{2} - 3\). That's what this becomes, \(3\sqrt{2}\). So again, they were unlike radicals first. Now I've simplified them, and they turn into like radicals. So now I can just go ahead and subtract them. \(5 - 3\) just becomes \(2\sqrt{2}\), and that's the answer. Alright, so what I want to do if you have a calculator handy is actually want you to plug this expression into your calculators. \(5\sqrt{2} - \sqrt{18}\). You're just going to get a number, and if you do \(2 \times \sqrt{2}\), you're going to get the exact same number. Alright. So, this is just another way, a simpler way to write that expression that we started with. Alright. That's the whole idea. Let's take a look at this last one over here. \(\sqrt{18}\) and \(\sqrt{50}\). Alright. Same idea. I can only add them when these two square roots are like each other. So here's the question. Can I take the square roots, and can I simplify it so that I get a \(\sqrt{50}\) out of it? Well, no, because if we break it down, all the numbers get smaller. But can I take the \(\sqrt{50}\) and break it down so that I get a \(\sqrt{18}\) out of it? Because then I would be able to add them. Well, let's try that. Let's try to break down this \(\sqrt{50}\) into a product of \(2\) terms, which you get an \(18\). So, basically, what you're asking is, is \(50\) divisible by \(18\)? Well, if you do \(18 \times 2\), that's \(36\). And if you do \(18 \times 3\), that's \(54\). So \(18\) doesn't evenly go into \(50\). So, in other words, I can't sort of break this thing up into a product where I get \(18\). So how do I solve this problem? Well, it turns out that unlike these sorts of problems over here, where we only had to break down one of the terms, in problems, you might actually have to simplify both of the terms before you can start combining them. So we're going to have to break down the \(18\) and the \(50\), and the hope is that you're going to get two radicals that are the same. Alright? So that's the idea. So we've actually already seen how to break up the \(\sqrt{18}\) in the other problem in part A. We saw that this just breaks down into \(\sqrt{9} \times \sqrt{2}\). What happens to the \(50\)? Well, if you go down the list, \(4\) doesn't go into \(50\), \(9\) doesn't either. \(16\) doesn't go into \(50\). What about \(25\)? \(25\) does go into \(50\). So in other words, it just becomes \(25 \times \sqrt{2}\). So now what you've seen here is by breaking down the \(\sqrt{18}\) and the \(\sqrt{50}\), when you factor it out, and you pull out perfect powers, basically you end up with the same radicals. Alright. So let's clean this up a little bit more. What does \(9\sqrt{2}\) become? That just becomes \(3\sqrt{2}\). And then what does the \(25\sqrt{2}\) become? That just becomes \(5\sqrt{2}\). Remember, we had an addition sign over here. So now I've basically ended up with two radicals that are alike, So now I can add them, and this whole thing just becomes \(8\sqrt{2}\). Alright. So again, if you have a calculator handy, go ahead and plug in this expression in your calculator. \(\sqrt{18} + \sqrt{50}\). You're going to get a number. I think it's something like, I think it's like \(11.7\) or something like that. And if you plug in \(8\sqrt{2}\), you're going to get that exact same number. Alright? So that's it for this, folks. Thanks for watching. I'll see you in the next one.
Here’s what students ask on this topic:
How do you simplify the square root of 20?
To simplify the square root of 20, you need to break it into a product of factors where one factor is a perfect square. The number 20 can be factored into 4 and 5, where 4 is a perfect square. So, you can write:
Therefore, the simplified form of the square root of 20 is 2√5.
Created using AIHow do you simplify the cube root of 54x4?
To simplify the cube root of 54x4, break it into a product of factors where one factor is a perfect cube. The number 54 can be factored into 27 and 2, where 27 is a perfect cube. For the variable part, x4 can be split into x3 and x:
Therefore, the simplified form is 3x√[3]{2x}.
Created using AIHow do you simplify the square root of a fraction like 49/64?
To simplify the square root of a fraction like 49/64, you can break it into the square root of the numerator and the square root of the denominator:
Therefore, the simplified form of the square root of 49/64 is 7/8.
Created using AIHow do you add and subtract radical expressions?
To add and subtract radical expressions, you need to combine like radicals, which have the same radicand and index. For example, to simplify 2√x + 4√x + 3:
Only like radicals can be combined. For example, 2√x and 4√x can be combined, but 2√x and 3√y cannot.
Created using AIHow do you simplify the square root of x3?
To simplify the square root of x3, break it into a product of factors where one factor is a perfect square. The exponent 3 can be split into 2 and 1:
Therefore, the simplified form of the square root of x3 is x√x.
Created using AI
