Simplifying Radical Expressions: Study with Video Lessons, Practice Problems & Examples

Understanding how to simplify radicals is essential in mathematics. When dealing with square roots or cube roots, expressions like $\sqrt{20}$ can be simplified by breaking them into products of perfect squares, such as $\sqrt{4}$ and $\sqrt{5}$. Additionally, combining like radicals requires that they share the same radicand and index, allowing for straightforward addition or subtraction, enhancing problem-solving skills in algebra.

Expanding Radicals

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, sort of like making them a little bit bigger so that we can hopefully make them 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 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 4 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 it'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√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 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 18 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 the 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 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√2 and that is our fully simplified expression.

Simplify the radical.

$\sqrt{75}$

• A. $3\sqrt5$
• B. $3\sqrt{25}$
• C. $5\sqrt3$
• D. $25\sqrt3$

Simplify the radical.

$\sqrt{180}$

• A. $6\sqrt5$
• B. $3\sqrt5$
• C. $3\sqrt{20}$
• D. $2\sqrt{45}$

Radical Expressions with Variables

Everyone, up until now, all the radicals that we've seen in our problems have always involved only numbers. So, 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 x3, or you might even see numbers and variables. What we're going to see in this video is that the way 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 because 2 times itself 3 times gives me 8. Opposites. The idea is the same. Imagine I had something like or if I had, you know, a radical x2. What that's asking me to do is, what is the thing that multiplies by itself to get me x2? And it actually turns out to just be x. If I just take x and multiply by itself, I get x2. So that means that the square root of x2 is just x. Alright?

Now let's say I have, like, the cube root of x6. What's something if I multiply it by itself 3 times that gets me x6? Is it x? Well, no. Because x3 is only going to give me x3, 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 basically just multiply their exponents and I get x6. So that means that the cube root of x6 is just x squared. Alright? So we can use basically the same things that we were doing with square and nth roots, and we can also just apply them to variables.

That's really all there is to it, so let's go ahead and 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 x3. This is a little bit different from the example that we worked up here where we had the square root of x2. Basically, I want to find something where, if I multiply it by itself, I get to x3. So let's try that. Let's see if I take x and square that. That's just x2. Well, what about x squared squared? Right? Can I use the power rule here? And if you work this out, what you're going to get is you're going to get x4, which is too big. Remember, we're trying to get x3, so we've gone too far. So none of these will work, and it's actually because this thing isn't a perfect square. We need something that's sort of in the middle here.

So how do we deal with this when we had numbers? Well, when radicals weren't perfect powers, remember what we did is we split them, and the whole hope was that one of the factors was going to be a perfect power. We look for things like 4 or 9 or 16 or something like that. Basically, we're going to do the exact same thing with the variables. I'm going to split this into 2 radicals, and I want one of them over here to be a perfect power. How do I split x3? Well, it’s basically sort of like undoing the power rule. How can I break up the exponent of 3? Well, I can do x2 and x. Now, why is this helpful? Because this x2, we already know what it turns out to be. This just turns out to be x. So this thing over here is x, and then what you're left with is you're left with 1 power of x over here. Okay? So this is the answer.

This is how you take radical x3 and simplify it. These 2 mean the same exact thing. I want to show you a cool trick that will always happen because it might be helpful. What happens is there's a hidden 2 that's an index, so you have to ask yourself, how many times does 2 go into 3? It goes in one time. So in other words, you do 1 times 2, and that's the exponent that you can pull out of x3. Let me show you again how it works with a little bit of a more complicated example.

So, again, here, if I wanted to do the square root of x7, rather than having to break it up a bunch of times, there's an index of 2. How many times does 2 go into 7? It goes in 3 times.

Simplify the radical.

$\sqrt{63x^2}$

• A. $63\sqrt{x}$
• B. $3\sqrt{7x}$
• C. $x\sqrt{63}$
• D. $3x\sqrt7$

Radical Expressions with Fractions

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 4964. Now this fraction over here, 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, 322 over 22. So if I want to do the same thing over here, 322 is really difficult for me to evaluate, and so is 22. 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 322, 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^49x^2. 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 to the 4th is also a perfect power. It's a perfect square. So this 64 is just 8 squared, and this x to the 4th power is just x squared squared. Right? So using the power rule. So, basically, what this whole thing actually just becomes, 64x to the 4th, is it just becomes x 8x2. That's the square root of that. What does 9x2 become? Well, 9 is a perfect square of 3 and the x so this is 3 squared, and the x squared is just a perfect square of x squared. 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 squared 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 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 radical 9, that is a perfect power, but x isn't. And 72 isn't a perfect power, and we also have this x cubed 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 to the 3rd 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 cubed over x? This is basically what we just did over here with the exponent rule. This actually just becomes x squared. 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 42, and the x squared 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

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. When we dealt with algebraic expressions, if I had something like \(2x + 3 + 8\), I simplified it by combining like terms. I could combine \(2x\) and \(4x\) and \(3\) and \(8\), and this just became \(6x + 11\). The idea is that when I have a radical expression, instead of combining like terms, I'm going to combine like radicals. Like radicals just means that they have the same radicand and the same index. I have to make sure that they're both square roots, not that one is a square root and one is a cube root. As long as we have the same radicand and the same index, we can just add them.

Here's how we add them: \(2 \sqrt{x} + 4 \sqrt{x}\) is similar to adding apples and apples. Right? So this just becomes \(6 \sqrt{x}\), and then the \(3\) and \(8\) just become \(11\) like they always have. This is how you simplify these kinds of expressions. You can only combine things that are like each other. That's all there is to it. Let's go ahead and take a look at a couple more examples here.

So if we have \(3 \sqrt{7} \times 2 + 2 \sqrt{7} - \sqrt[3]{7}\), how do I simplify and this expression? I can only combine like radicals. Like radicals have like radicands and the same index. They have the same thing under the radical and the same index. Here, we'll notice that all the radicands are \(7\). However, the indexes are not the same. Here I have an index of 3, and here I have square roots, which are indices of 2. So the \(3\) and \(2\) combine down to \(5 \sqrt{7}\), and then I have minus \(\sqrt[3]{7}\). This is like an apple, and the cube root of 7 is like a banana. I can't add those because they're not the same. And that's the answer.

Pretty straightforward. However, I want to warn you against a common mistake. Whenever you see radicals that are separate from each other and you're adding them, you can't combine them into one radical. For example, \( \sqrt{7} + \sqrt{7} \) is not equal to \( \sqrt{14} \). This is a mistake I see a lot of students make. Just be very careful that you don't do this. Otherwise, you're going to get the wrong answer. \( \sqrt{7} + 7 \) does not equal \( \sqrt{14} \). You can't just merge those into the same radical.

We have another example here. Here, we have \(9 \sqrt[3]{x}\), then we have \(4 \sqrt{3} \cdot x\). I've got the same radicands. I've got \(x\) everywhere, but I have different indexes. Here, I have a cube root of \(x\) and a square root of \(x\). So, what can I combine? I can combine the 9 and the 4, keeping the sign over here. This becomes \(5 \sqrt[3]{x}\), then I have minus \(\sqrt{x}\), then plus 3. That's how you simplify this expression. That's all there is to it, folks. Let me know if you have any questions, and thanks for watching.

True or False:

$\sqrt{9+16}$ and $\sqrt9+\sqrt{16}$ are equal.

• A. True
• B. False
• C. Cannot be determined

Adding & Subtracting Unlike Radicals by Simplifying

Everyone, so earlier in videos, we saw how to add and subtract like radicals. For example, I could take this 3√5 and 4√5, I have the same number in the same index, so I can combine them. So this 3 and 4 just becomes 7. You just add the little numbers in front, and I have 7√5, and that was the answer. What I'm going to show you in this video is that sometimes you'll actually have radicals and square roots that aren't like each other, like √5 and √20, and you're going to have to add them. How do we solve this? I'm actually going to show you in this video how we do that, and we're going to use some old ideas that we've already seen before. Let's go ahead and get started. Basically, the idea is that when you're adding subtracting radicals that aren't like each other, you're going to have to simplify them first. So simplify before you end up combining their like radicals. So here's the idea. Can I simplify the √5? In other words, can I pull out a perfect power? Well, no. Because the only things that factor into 1 and 5 are 1×5. 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 how do we do this? Well, 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 I can split it up into the product of √4 and √5. So why is this helpful? Well, because this just becomes √5 plus the √4 just becomes 2, and this whole thing just becomes 2√5. So in other words, I've ended up or I've started with 2 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 are like radicals. And this basically just turns into the problem on the left. So how do I add these things? Well, √5 plus 2√5. You just add the numbers in front. This is just 3√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. So here, we're going to do 5√2 and √18. Alright? So what happens is can I simplify the √2? Well, no. Well, the whole idea is that I want these 2 square roots to eventually be the same because then I can subtract them. So I can't simplify the √2, but can I simplify the radical 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 in this case. So I'm going to break this up into a product of 2 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√2 minus 3. That's what this becomes, 3√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 just subtract them. 5–3 just becomes 2√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√2 minus √18. You're just going to get a number, and if you do 2 times √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. √18 and √50. Alright. Same idea. I can only add them when these two square roots are like each other. So here's a question. Can I take the square roots, and can I simplify it so that I get a square root of 50 out of it? Well, no. Because if we break it down all the numbers get smaller. But can I take the √50 and break it down so that I get a √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 √50 into a product of 2 terms, which 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 sort 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 2 radicals that are the same. Alright? So that's the idea. So we've actually already seen how to break up the √18 in the other problem in part a. We saw that this just breaks down into √9 times √2. What happens to the 50? Well if you go down the list, 4 doesn't go into 50, 9 doesn't either. The 16, 16 doesn't go into 50. What about 25? 25 does go into 50. So in other words, it just becomes 25 times √2. So now what you've seen here is by breaking down the √18 and the √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√2 become? That just becomes 3√2. And then what does the 25√2 become? That just becomes 5√2. Remember, we had an addition sign over here. So now I've basically ended up with 2 radicals that are alike. So now I can add them, and this whole thing just becomes 8√2. Alright. So again, if you have a calculator handy, go ahead and plug in this expression in your calculator. √18 plus √50. You're going to get a number. I think it's something like 11.7 or something like that. And if you plug in 8√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 down into a product of perfect squares. The number 20 can be factored into 4 and 5, where 4 is a perfect square. So, you can write:

$\sqrt{20}=\sqrt{4}*\sqrt{5}=2*\sqrt{5}=2\sqrt{5}$

Thus, the simplified form of the square root of 20 is 2√5.

How do you simplify the cube root of 54x⁴?

To simplify the cube root of 54x⁴, break it into two parts: the cube root of 54 and the cube root of x⁴. First, factor 54 into 27 and 2, where 27 is a perfect cube:

$\sqrt[3]{54}=\sqrt[3]{27}*\sqrt[3]{2}=3*\sqrt[3]{2}$

Next, for x⁴, split it into x³ and x:

$\sqrt[3]{x^4}=\sqrt[3]{x^3}*\sqrt[3]{x}=x*\sqrt[3]{x}$

Combining these, the simplified form is:

$3x\sqrt[x]{2}$

How 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:

First, identify like radicals. Here, 2√x and 4√x are like radicals because they have the same radicand (x) and index (2). Combine them:

$2\sqrt{x}+4\sqrt{x}=6\sqrt{x}$

Then, add the constant term:

$6\sqrt{x}+3=6\sqrt{x}+3$

So, the simplified expression is 6√x + 3.

How do you simplify radical expressions with variables?

To simplify radical expressions with variables, apply the same rules as with numbers. For example, to simplify √(x³):

First, break it into parts where one part is a perfect square:

$\sqrt{x^3}=\sqrt{x^2*\sqrt{x}=x*\sqrt{x}}$

So, the simplified form is x√x.

How do you simplify radicals with fractions?

To simplify radicals with fractions, you can split the fraction into separate radicals for the numerator and the denominator. For example, to simplify √(49/64):

First, split the fraction:

$\sqrt{\frac{49}{64}}=\frac{\sqrt{49}}{\sqrt{64}}$

Then, simplify each part:

$\frac{7}{8}$

So, the simplified form is 7/8.