Simplifying Radical Expressions - Video Tutorials & Practice Problems

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1

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Expanding Radicals

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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 nine, which is three or the cube roots of eight, which is two. But in a lot of problems, that's not gonna happen in a lot of problems, you might see something like the square of 20 or the cube root of 54. And you're gonna have to know how to take those expressions and make them simpler. That's what I'm gonna 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 is sort of like making them a little bit bigger so that we can hopefully make it smaller and simpler. Later on, I'm gonna 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 20 the whole thing we're gonna do is we're gonna simplify it by turning it into a product, we're gonna try to break it up into a product of two things and the whole goal is that one of the, the terms will be a perfect power. So here's the thing, I'm gonna take the radical 20 I'm gonna break it up so that it's the product of two things. And I want one of these things to be a perfect power like four or nine or 16 or something like that, right? So I have this table here. We're gonna just go down this table and see if we can turn the 20 into a product where four or nine is one of the terms. So can this happen? So can I do four? Well, if you take a look at 44 times five, that equals 20. So basically, you just separated this thing into two radicals and I can totally do that. So what's the square root of four? We have already seen that's just two. What's the square root of five? Well, that's just the square root of five and that's not a perfect number. So basically, what we've seen here is that we've turned this into a two times radical five. And so the simplest way that we can rewrite this expression is just two radical five. Now, can we go any further? No, because five 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. All right? But that's the basic idea. So as a formula, the way that you're gonna see this in your text because if a, 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. All right, that's the whole thing. Let's move on to the second problem now because in some problems, you're gonna have variables as well. So what I like to do is I like to separate this thing into the number times the variable. All right. So again, when I take these radicals, can I break them up into anything in which we're gonna get a perfect square out of it? Well, let's do the 18 1st. So it does, does 18 reduce to anything. Uh So does four go into 18. Well, four times four is 16, 4 times five is 20. So it doesn't what about nine? Well, actually 18 can be written as the product of radical nine times radical two, 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. All right. Now, are we done yet? Is this our full expression? Well, no, because the squared of nine actually just turns into a three. So uh what about the radical two? Can we break up the radical two any further? No, because it's just a prime number. And so basically, what happens is this is our simplest, so we can write this expression. And what you're gonna see here is that the X usually gets moved in front of the radical. So this whole thing really just becomes three X radical two. And that is our fully simplified expression. Now, for the last one over here, we have the Q root of 54 X to the fourth power. So now we no longer have square roots, we have cube roots. But the idea is the same. Uh and again, what I like to do here is break this up into two radical. So radical 54 C 54 and then the cube root of X to the fourth power. And I'll just deal with those independently, right? So let's do the 54 1st. Can I break this up into a perfect cube? Uh Perfect cubes are gonna be over here like eight and 27 and stuff like that. So what about 88 doesn't go into 54 8 times seven is 56. So it's close. What about 2727? Um Let's see, 27 is actually, yeah, this does work. So in other words, this is just the cube root of 27 times the cube root of two. So I've gotten a perfect cue out of this. All right. Now, what about the cube root of X to the fourth power? Is there a perfect cube that I can pull out of that? Well, think about it. This is just X multiplied by itself four 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 on do it right. So with this 27 the Q root of this turns turns out to be is just three. Now, the Q root of two doesn't simplify. But what about the QE 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 Qbe root of X over here is just left alone. All right. So how do I make this simpler? Well, I just mash the three and the X together and just becomes three X. 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 uh cube roots of uh this is becomes two X over here. And this is the fully simplified expression. I can't break this up any further. All right. So that's all there is to it. Folks. Let me know if you have any questions. Let's get some practice.

2

Problem

Problem

Simplify the radical.

$\sqrt{75}$

A

$3\sqrt5$

B

$3\sqrt{25}$

C

$5\sqrt3$

D

$25\sqrt3$

3

Problem

Problem

Simplify the radical.

$\sqrt{180}$

A

$6\sqrt5$

B

$3\sqrt5$

C

$3\sqrt{20}$

D

$2\sqrt{45}$

4

concept

Radical Expressions with Variables

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6m

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Everyone. So up until now, all the radicals that we've seen in our problems have always just involved only numbers. So for example, like the square root of nine or the cube root of eights, but in some problems, you actually 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 very we're gonna see in this video is that the way that we handle radicals with variables is actually exactly how we've been doing it with just numbers. We can apply everything that we've seen from square roots and roots and even simplifying whenever we have variables in radicals, let's go ahead and get started. We'll do a couple of examples together. So when we had radicals without variables, like just numbers, remember that the square root of nine was three because three squared was equal to nine. So these were basically the opposites of each other. So when I had the Q root of eight, that's a number multiplied by itself three times that gets me eight and that was two. That's because two times itself three times gives me eight opposites 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 actually turns out to just be X if I just take X and, and multiply by itself, I get X squared. So that means that the square root of X squared is just X, all right. That kind of makes sense. One of the things that we saw in an earlier video is if you have a term that's raised to an exponent and the exponent is the same as the index, you're basically just canceling each other out, right? So if I take a number and then square it and then square root it, I'm kind of just like going forwards and backwards. So this just turns out to be just X. All right. Now, let's say I have like the cube root of X to the sixth power. What's something if I multiply it by itself three times that gets me X to the sixth is it X? Well, no, because X cubed is only gonna be X cubed. So that doesn't work. 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 X to the six. So that means that the cub root of X to the sixth power is just X square. All right. So we can use basically the exact same things that we were doing with square and N roots, but we can also just apply them to variables. All right. That's really all there is to it. So let's just 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 X cubed. This is a little bit different from the example that we worked up here where we had square root of X squared. Basically, I wanna find some thing where if I multiply it by itself, I get to execute. So let's try that. So let's just try, let's say if I take X and square, that's, that's just X squared. Well, what about X squared squared? Right? Can I use the power rule here? And if you work this out, what you're gonna get is you're gonna get X to the fourth power, which is too big. Remember we're trying to get X to the third power. 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 I hoped was that one of the factors was going to be a perfect power. We look for things like four or nine or 16 or something like that. Basically, we're gonna do the exact same thing with, with the variables. I'm gonna split this into two radicals and I want one them over here to be a perfect power. How do I split X cubed? Well, it's basically sort of like undoing the power rule. How can I break up the exponent of three? Well, I can do two and one. So in other words, I can do radical of X squared and the radical of X. Now, why is this helpful? Because this radical X squared, 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 one power of X over here. OK. So this is the answer, this is how you take radical X cubed and simplify it. These two mean the same exact thing I actually wanna show you a cool trick that will always happen because it might be helpful. Uh What would you if you remember here, what happens is there's sort of like a hidden two, that's an index. So you have to ask yourself how many times does two go into three? It goes in one time. So in other words, you do one times two and that's the exponent that you can pull out of that uh X to the third power. Let me show you again how it works with a little bit of a more complicated example. So a again here, if I wanted to do the square root of X to the seventh power, rather than having to break it up a bunch of times, there's an index of two. How many times does two go into seven? It goes in three times. So what you can do here is when you write this out and you splitt out your two radicals, then this turned out to be X cubed. So in other words, I take the three times and that becomes my exponents and I'm gonna raise it to the second power. So and then uh over here, what are you left with? Well, I have six powers of X that are covered here. Let me just write this a little bit bigger. So I have the square root of X cubed squared. I have six powers that are covered over here. Whereas I have seven powers that I started out with. So what gets left over here? Just one power of X? Now, why is this helpful? Because when you take the square of a square root basically just cancel each other out. And all you're left with here is just X cubed times radical X. So it's a really cool shortcut to be able to take really high power of X and actually just work them down pretty quickly. All right. So that is how you simplify this. So now for our last question here, we actually have numbers and we have uh XS and variables. Basically, the way this works here is it's no difference. If you have radicals with numbers and variables, you can actually just split them out into their own separate radicals, kind of like how we did with XS and then you can simplify them separately. So the whole idea here is that I can take the eight and the X of the fifth split them out into their own radicals because I know how to simplify radical eights. Um And we'll just deal with these things separately. All right. So how do we deal with uh radical eight? Well, remember we split this out into radical two times or sorry radical four times, radical two radical four is a perfect square over here. So this just turns out to be two times radical two. Now, what is the X to the fifth turn out to be? Well, we can use our trick or shortcut. There's an index of two. How many times does two go into five? It goes in twice. Um And so what we do here is we just pull out a factor of um this is gonna be X squared to the second power squared. And then we have four powers of X that are covered here. What's left over one power of X. So what happens is the square root T cancels and all you're left with over here is the square roots times or sorry, X squared times the square root of X. So now what we have to do is we have to take all of this stuff and we have to just put it back together again because all of these things um are, are the, are things that multiply the A X to the fifth. So I'm gonna take the X squared, I'm gonna put over here, this just becomes two X squared and I have a radical two and then a radical X. So I could basically just put them back together again and under the same radical. So I have this. So if you take this and you multiply it out by itself, you should get back to eight X to the fifth power. So that is your final answer. Anyway, folks, thanks for watching. That's it for this one.

5

Problem

Problem

Simplify the radical.

$\sqrt{63x^2}$

A

$63\sqrt{x}$

B

$3\sqrt{7x}$

C

$x\sqrt{63}$

D

$3x\sqrt7$

6

concept

Radical Expressions with Fractions

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5m

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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, and 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 it. I'm gonna 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, let me show you how to, 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 two different things with radicals. We can take a fraction and break it up into two 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 49/64. Now, this fraction over here, 49/64. If I want to divide this first, it would be really difficult because I'm gonna 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 could just break up this radical, you can break up this radical in to square to 49 and square to 64. And these are actually very easy. Squared to 49 is just seven and squared of 64 is just eight. All right, let's take a look at another example here, radical 32 over radical two. So if I want to do the same thing over here, radical 32 is really difficult for me to evaluate and so is radical two, those are not perfect squares. However, what I can do here is I actually can sort of combine them into one radical and say and say this is a radical 32/2 and this just becomes radical 16 and radical 16 is a perfect square that just evaluates the four. 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 uh example over here. And then sometimes it's actually better to divide first and then you simplify over here. All right. All right. 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 uh let's take a look at this first one X 64 X to the fourth power over nine X squared, all of that underneath one radical. So if we try to divide this first with what's gonna happen is 64/9 isn't going to give me a clean number. But I do notice that 64 and nine are both perfect squares. So let's try to break them up into their own radicals. This just becomes radical 64 X to the fourth divided by radical nine X squared. All right. So 64 is a perfect power and X to the fourth is also a perfect power. It's a perfect square. So this 64 is just eight squared and this X to the fourth power is just X squared at squared, right. So using the power rule, so basically what this whole thing actually just becomes 64 X to the fourth is it just becomes X eight X squared. That's the square root of that. What is nine X squared become? Well, the nine is a perfect square of three and the X, so this is three squared and the X squared is just a perfect square of X squared. So in other words, this whole thing actually just becomes three X 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 um like eight X squared and three X 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 be really just becomes an exponents problem. Basically, I'm gonna use the quotient rule and what this answer becomes is it just becomes eight over three X, all right. Basically, just delete one power of X on the top and the bottom and all you're left with is 8/3 X, all right. That that's all there's to it. Now, let's look at the second one here, 72 divided by nine. If I try to do this and try to sort of treat them as independent. What happens is the radical nine 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 gonna be kind of tricky to sort of separate this and deal with them themselves. So let's just try to combine them all under one radical and see what happens. So I'm gonna combine this as so 72 X to the third over nine X. And then basically what happens is we're gonna do the division first before we actually do the radical. So what is 72/9? This just becomes eight. 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 radical eight X squared. Can we simplify this? Well, if you notice that eight can be broken up into a perfect square because eight factors into four and two 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 a radical eight times radical X squared. And we'll deal with those separately radical eight just becomes radical four times radical two. We saw that from the previous video and then this radical X squared actually just factors out into one power of X. This is a perfect square. So this just simplifies 22. we can move the X to the front and this just becomes two X radical two. Now, is this fully simplified? Yes, because we can't uh we can't factor anything else out. So this is basically what this whole expression becomes. All right. So that's it. For this one. Folks, let me know if you have any questions, see in the next one.

7

concept

Adding & Subtracting Like Radicals

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3m

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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 gonna go over in this video. Um It turns out that it's actually very similar to how we dealt with algebraic expressions. And I'm gonna 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 two X plus three and four X plus eight, the way I simplified this was I combined like terms, I could combine the two X and the four X and the three and the eights and this just became six X plus 11. The idea is that when I have a radical expression um instead of combining like terms, I'm gonna combine like radicals. So like radicals just means that they have the same radic hand, 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 cri or something like that. As long as we have the same Ratican and the same index, we can just add them and how do we add them? We add them exactly how we added two X and four X. So two times the square root of X and four times the square root of X. It's kind of like I'm adding apples and apples, right? So this just becomes six times the square root of X. So that's six root X and then the three in the eight just become 11, how they like just like they always have. So this is how you simplify this kind of expressions. You can only combine things that are like each other, right? That's all there is to it. So let's go ahead and take a look at a couple more examples here. All right. So if we have, let's say three radical seven times or sorry, three square of seven times two plus two square root of seven minus the cube root of seven, how do I simplify and add this expression? Well, remember I can only combine the like radicals like radicals have like radic hands and the same index and the same thing under the radical and the same index here. What we have, you'll notice that all of the radic hands are seven, I have sevens in all of the symbols over here. But all the index is the same. Well, no, because here I have an index of three whereas here I have square roots and remember those are indexes of two. So these two things have the same Ratican and the same index. But this one is, has the same Ratican but a different index. So it's not a like term. OK. So just be very careful when you're doing that. So basically what happens is I can combine the two, the two things in yellow. So the three and the two just be just combined down to five radical seven or square root of seven. And then I have over here minus the cube roots of seven. So this is like an apple and the cube root of seven is like a banana. I can't add those things because they're not the same. And so this is my how I um this is how I simplify my expression and that's the answer. All right. So pretty straightforward. Now, what I want to warn you against actually is something that I see a lot of students uh get this, you know, make this mistake. Um Basically, when you're pursue radicals that are, you know, separate from each other, um and you're adding them, you can't combine them into one radical. So for example, I can't take the square root of seven and the square root of seven that's not equal to the square root of 14. This is a, this, 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 seven plus radical seven does not equal radical 14. You can't just like merge that stuff into the same radical. All right, just be very careful. All right. So let's look at another example here, here we have nine 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 radicans, I've got X's everywhere. Um But I have different indexes here. What I have is I have a Q root of X and a Q root of X. And then here I have a square root of X. So those are different, they're not like radicals and then I just have a constant over here. So what can I combine? I can combine the nine and the 49 minus four? Remember just keep, keep the sign over here. This becomes four times the key root of X. Then I have minus radical X, then I have plus three. So that's how to simplify this expression. All right. So that's all there is to it, folks. Uh Let me know if you have any questions and thanks for watching.

8

Problem

Problem

True or False:

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

A

True

B

False

C

Cannot be determined

9

concept

Adding & Subtracting Unlike Radicals by Simplifying

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5m

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Everyone. So earlier in videos, we saw how to add and subtract like radicals. For example, I could take this three square to five and four square to five have the same number and the same index so I can combine them. So this three and four just becomes seven, you just add the little numbers in front and I have seven square to five. And that was the answer. What I'm gonna show you in this video is that sometimes you'll actually have radicals and square roots that aren't like each other like square to five and square of 20 you're gonna have to add them. How do we solve this? I'm actually gonna show you in this video how we do that. And we're gonna 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 gonna have to simplify them first. So simplify before you end up combining the like radicals. So here's the idea. Can I simplify the square to five. In other words, can I pull out a perfect power? Well, no, because the only things that were, you know, factor into one in uh five or one in five. Can we take the 20 break that down into a perfect power? Well, actually, yes, we can because we saw how to do this. I'm gonna take the 20 and I'm gonna break this down into a product. And the hope is that one of the uh radicals ends up being a perfect power. So, how do we do this? Well, we looked at 49 and 16 all the way up to 20 we saw that 16 and nine didn't work because they don't go into 20 but four does. So in other words, I can take this radical 20 I can split it up into the product of square to four and square to five. So why is this helpful? Well, because this just becomes square to five plus the, the square to four just becomes two and this whole thing just becomes two square to five. So in other words, I've ended up or, or I've 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 radic hands and 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, square to five plus two square to five, you just add the numbers in front. This is just three square to five and that's how to solve these kinds of problems, right? 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 gonna do five squared of two and five and squared of 18. All right. So what happens is, can I square, can I simplify the squared of two? 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 the square root of two. But can I simplify the minus or, or the radical 18? And we saw how we can do that by pulling out a perfect square. In this case, four isn't going to work but nine will in this case. So I'm gonna bring this up into a product of two radicals, nine goes into 18 and I get two as what's left over. So again, what we end up with here is we end up with five squared of two minus three. That's what this becomes three squared of two. 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 five minus three just becomes two squared of two and that's the answer. All right. So what I want you to do in, if you have a calculator handy is actually want you to plug this expression into your calculators, five squared of two minus the squared of 18. You're just gonna get a number and if you do two times a squared of two, you're gonna get the exact same number. All right. So this is just another way, a simpler way to ex to write that expression that we started with. All right, that's the whole idea. Let's take a look at this last one over here, squared of 18 and squared of 50. All right, 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 to 50 out of it? Well, no, because if we break it down, all the numbers get smaller, but can I take the square to 50 break it down so that I get a square of 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 square root of 50 into a product of two terms which I get an 18. So basically what you're asking is, is 50 divisions by 18. Well, if you do 18 times two, that's 36. And if you do 18 times three, that's 54. So 18 doesn't divis 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 and problems, you might actually have to simplify both of the terms before you can start combining them. So we're gonna have to break down the 18 and the 50. And the hope is that you're gonna get two radicals that are the same. All right. So that's the idea. So we've actually already seen how to break up the squared of 18 in the other problem. In part A we saw that this just breaks down into squared of nine times squared of two. What happens to the 50? Well, if you go down the list four, doesn't go into 59 doesn't either. The 1616 doesn't go into 50. What about 25? 25 does go into 50? So in other words, just becomes 25 times the squared of two. So now what you've seen here is by breaking down the square root of 18 and the squared of 50 when you factor it out and you pull out perfect powers. Basically, you end up with the same radicals. All right. So let's clean this up a little bit more. What does nine square root two become that just becomes three square root of two? And then what does the 20 squared of 25 squared of two become that just becomes five squared of two? I remember we had an addition sign over here. So now I've basically ended up with two radicals that are like, so now I can add them and this whole thing just becomes eight squared of two. All right. So again, if you have a calculator handy, go ahead and plug in this expression of your calculator squared of 18 plus squared 50. You're gonna get a number. I think it's something like, like, uh I think it's like 11.7 or something like that. And if you plug in eight squared of two, you're gonna get that exact same number. All right. So that's it for this folks. Thanks for watching. I'll see you in the next one.