Simplify the following. 3 ( 2 − 6 ) \sqrt3(2-\sqrt6)

Folks. So let's keep moving on to our next practice problem here. We're going to simplify the following, but this one's a little bit different because not only do we have radicals involved, we also have a subtraction symbol. And remember, that just means that we're going to have to use the distributive property. Let's check it out. All that really means is we take the radical three and we distribute it to the two and also to the radical six that's inside the parentheses here. And we're going to end up with two different terms. When you multiply radical three times two, all you end up with is just two radical three, and you can't really do anything with that. That doesn't simplify. But then what happens is that you have radical three, so you have a minus sign over here, and then you have radical three times radical six. Alright? So now again, because of the product rule, you can combine these under one radical, and here's what you get. Here we get two radical three, and then you get a minus sign, and this becomes radical 18. Alright. Now, again, this two radical three doesn't simplify. There's no other way you can make this any simpler. Whenever you see large numbers inside of radicals, I always want you to try to think about two numbers that can multiply to that number in which one of them is a perfect square so that you can simplify that radical. All right. So let's list this out for 18. We've got one and eighteen, but that doesn't really change anything. But what about two and nine? Two and nine? Well, nine is a perfect square. So that's one way we're going to be able to simplify this radical here. All right. So here's what I want to do. I want to basically rewrite this so that's two radical three minus. And And then what I've got here is this is just the same as the product of two times nine. Now, what happens here is that the radical nine over here can actually be simplified further. Remember that just because just as you can combine two things over one radical, you can also split them up into separate radicals. So basically this radical 18 can become radical two times radical nine. It's the same exact thing over here. So how does this simplify? Well, what happens is the two radical three stays and then we have a minus and then radical two doesn't simplify any further. But then there's radical nine over here. The square root of nine is three. Now, normally, again, what happens is we're going to put the whole numbers in front of the radicals. So this just becomes three radical two. All right. Now what happens here is you might be tempted to try and subtract these two. But remember, we have a radical three in the first term and a radical two in the second. So because they're not the same radical, you can't go ahead and add and subtract these terms. And And this is the most simplified that this would be. All right? Again, so you first do the Distributive Property and then once you distribute anything that's out here to all the terms that are inside, then just go ahead and use your product rules and see if you can simplify any further. That's all for that's all for this one. Let me know if you have any questions and I'll see you in the next

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Beginning & Intermediate Algebra

11. Roots, Radicals, and Complex Numbers

Multiplying, Dividing, and Rationalizing Radicals

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Multiple Choice

Simplify the following.
$\sqrt{3} (2 - \sqrt{6})$

$2 \sqrt{3} - 3$

$3 \sqrt{2} - 2 \sqrt{3}$

$2 \sqrt{3} - 3 \sqrt{2}$

$3 - 2 \sqrt{3}$

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Verified step by step guidance

Start with the expression $\sqrt{3}(2 - \sqrt{6})$. Our goal is to simplify this by distributing $\sqrt{3}$ to both terms inside the parentheses.

Distribute $\sqrt{3}$: multiply $\sqrt{3}$ by 2, and then multiply $\sqrt{3}$ by $-\sqrt{6}$. This gives us $2\sqrt{3} - \sqrt{3} \times \sqrt{6}$.

Simplify the product $\sqrt{3} \times \sqrt{6}$. Recall that $\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}$, so this becomes $\sqrt{18}$.

Simplify $\sqrt{18}$ by factoring it into $\sqrt{9 \times 2}$, which equals $\sqrt{9} \times \sqrt{2} = 3\sqrt{2}$. Substitute this back into the expression.

Rewrite the expression as $2\sqrt{3} - 3\sqrt{2}$. This is the simplified form of the original expression.

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Struggling with Beginning & Intermediate Algebra?

Simplify the following.3(2−6)\(\sqrt\)3(2-\(\sqrt\)6)

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Simplify the following.
$\sqrt{3} (2 - \sqrt{6})$