Simplify the radical.180\sqrt{180}180

6 5 6\sqrt5 6 5 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z">

Simplify the radical. 180 \sqrt{180} 180 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z">

Hey, everyone. So in this practice problem, we're gonna simplify this radical, the square root of a 180. So let's take a look here. Now a 180 is a pretty big number. It's bigger than a lot of the numbers that we've seen so far. You'll also notice that 180 isn't a perfect square. The closest actually is, like, 13 or 14 squared, which is a 169 or 196, but it's definitely not a perfect square. So the whole idea is I actually want to break this up into a product in which the hope is that one of these things will become a perfect square. Alright? So let's try this out. Now you may look at this sort of table here, and you may not know what's the largest square root that you can pull out of the 180. It's actually really difficult because I may not know if 16 goes into a 180, and I don't think it does. Either just 25. I don't know about 36. I don't think 49 does either. So the whole thing is that they're actually it's hard to figure out sometimes what's the biggest thing you can pull it out what you can pull out. So the best way to handle this actually is just to start small and sort of chip away at it. So let me show you what this means here. Does does 4, my smallest square, does that go into 180? And actually, it does because 4 goes into 20 and 20 goes into 180, so it definitely goes in. So if I just do 4, do I have to multiply 4 by it to get to 180? Other words, what is 180 divided by 4? It's 45. So I had I get the product of 445, and now what I can do is this simplifies down to just 2 times the square root of 45. Now I'm not quite done yet because this 45 isn't as simple as it could be. Right? There's definitely other perfect powers that I can pull out. So here's what happens. Right? This 2, now what I can do with this 45 is I can just basically just continue chipping away at it. Right? I can split this up into a product in which one of these numbers here is gonna be a perfect power. 4 isn't gonna work, but 9 actually does. This becomes 9 times 5. So now what happens is I get 2, and really this just turns into a 3. So this is 2 times 3 radical 5 or square root 5. And And now what happens is if you multiply the 2 and the 3, this really just becomes 6 times the square root of 5. And this is simple as simple as it possibly could be because I can't pull out any more perfect powers out of 5. So this is my final answer. What I'm actually gonna show you here real quick, so, you know, you could move on from this video if you want, I'm gonna show you another way you could have done this because you may have also noticed that 9 goes into a 180. So I'll just show you another way you could have done this. Right? You could have also said, well, this could actually could be 9 square root of 9 times the square root of 20. 9 times 20 equals 180. So how does this work out? Right? And this basically just would just work out to 3 times square root of 20. Same idea. The 20 can still be simplified further, and we've actually seen how 20 simplifies down. This really just becomes 3, and then I have what I've over here, I have square root of 4 times square root of 5. We saw that from a previous video. And so what happens here, this just becomes 3 times 2 square root 5, and you're basically just getting the opposite of what you got over here. So this could have just been 6 times radical 5. There's actually even a third option. There is actually one of these numbers here that is big enough that goes into 180, and actually it's the 36. So the 3rd method you could have done is you actually could have just split this up into 36 times the square root of 5, and you would have just immediately got to your answer of 6 times the square root of 5. So if you don't know what the largest thing you can pull out is, you can basically just chip away at it, but there's so many different ways to get to the same answer when you have big numbers like this. So don't be afraid, just try a bunch of stuff out, and you'll usually get the right answer. Thanks for watching.

Simplify the radical. 180 \sqrt{180} 180 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z">

6 5 6\sqrt5 6 5 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z">

3 5 3\sqrt5 3 5 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z">

3 20 3\sqrt{20} 3 20 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z">

2 45 2\sqrt{45} 2 45 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z">

0. Fundamental Concepts of Algebra

Simplifying Radical Expressions

Precalculus

0. Fundamental Concepts of Algebra

Simplifying Radical Expressions

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Multiple Choice

Simplify the radical.
$\sqrt{180}$

$6\sqrt5$

$3\sqrt5$

$3\sqrt{20}$

$2\sqrt{45}$

Verified step by step guidance

First, identify the number under the radical: $ \sqrt{180} $. We need to simplify this expression.

Factor 180 into its prime factors: 180 = 2^2 \times 3^2 \times 5.

Use the property of square roots that $ \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} $. Apply this to the prime factorization: $ \sqrt{180} = \sqrt{2^2 \times 3^2 \times 5} $.

Simplify the expression by taking the square root of the perfect squares: $ \sqrt{2^2} = 2 $ and $ \sqrt{3^2} = 3 $. Therefore, $ \sqrt{180} = 2 \times 3 \times \sqrt{5} = 6\sqrt{5} $.

Multiply the simplified radical by the coefficient outside the radical: 180 \times 6\sqrt{5} = 1080\sqrt{5}. Now, compare this with the given options to find the correct answer.

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