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In $3-41$ , express each product in simplest form. Variables in the radicand with an even index are non-negative.$$\sqrt{12 x y^{3}}(\sqrt{3 x y}+3)$$

$6 y(x y+\sqrt{3 x y})$

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Chapter 3

REAL NUMBERS AND RADICALS

Section 5

Multiplying Radicals

Whole which of Numbers

Fractions and Mixed Numbers

Decimals

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In $3-41$ , express each p…

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in this problem. For the square root of 12 x y cube, touch the quantity of the root of three x Y plus three. So to start, we're going to distribute, which will leave us with root. 12 x y cubed times, Route three X y plus three, Route 12 x y cubed. This three is out in front just because that's the way that we normally right coefficients, they go out in front. Next, we need to combine these two into the same radical sign so this will be equal to the square root of 12 x y Cubed times three x y plus three Route 12 x Y cute. Moving on into another line. We're going to combine like terms inside this radicals were to do the 12 times three we're gonna do X times x we're gonna do Why cubed times? Why? Which is why the first. So this all is equal to the square root of 12 times. Three is 36 x times x is X squared and why cubed times wider. The first is why to the force this all is plus the cube or sorry three, Route 12 X Y cute, continuing to simplify. We can break apart this into multiple pieces, so I'm gonna rewrite it as separate radicals. This is the square root of 36 times the square root of X squared times the square root of wide. In the fourth plus three, Route 12 X y Cube Route 36 is just six. The square root of X squared is X and the scared of why the fourth is why square So we have six x y squared. Plus, we now need to break apart this part on the right, so this will be three times the square root of 12 times the square root of X Y cube. Breaking this down even further. This Route 12 can be simplified. We need to think what to numbers multiplied together, lead to 12 and we can use three and four. We're not going to use six and two because neither six nor to our perfect squares in three and 44 is a perfect square. I'd much rather used that. This will be six x y squared plus three times Route four times. Route three This route four times 33 Coming from that route 12 This all is times root X Y cubed. Now Route four is just too. So I'm going to rewrite again as six x y squared plus three times two times route three times Route X Why cubed and this three times to congest be rewritten as six. So that's what I'm gonna do right here. I'll just erase and rewrite as the number six. So now we have six x y squared plus six times route three times root X Y huge and we're gonna continue to solve. So it looks like over here I never multiplied the three and two together, So I'll do that too. All Replace that with a six So six x y squared plus six word three times root x y cubed up Next we can simplify even more. This part right here If we have inside a square root Why cubed? This is the same thing as the square root of why times Why squared, which is the same thing, is the squared of why times square root of y squared, which is the same thing. Is the square root of why times Why? Because spirit of anything squared. It's just the number on the inside. So this part right here. This why is going to get replaced? So I'm gonna go ahead and rewrite that, using this knowledge that we have right now that we can replace the square root of why cubed with why times the square root of why So this will be six x y squared plus six route three times the square root of X times the square root of why cubed, moving forward and using that knowledge that we established over here, I'm going to continue rewriting. This is six x y squared plus six route three times the square root of X times. Why times the square root of Why again Coming from over here. For our final answer, we need to rearrange all of these parts and separate out anything that all the terms have in common. So I'm going to combine like terms. First of all, I'm gonna combine this three x and why which are all in the routes into one big route. And I'm going to multiply together the six and the why putting them as the coefficient. So when I rewrite this, this all is equal to six x y squared plus six times why which is just six. Why times the square root of all of these parts in the roots combined since they were all multiplied together Route three X Why? And finally we need to pull out anything that's in common. And I noticed that we have a six and a why here. And I noticed we have a six. And why here? So that's gonna be the coefficient out in front. I'm gonna put out a six and a why. And I'm going to divide six y from both of thes terms. So if I take six x y squared and I remove, I divide out a six. Why? I will have one x right because six divided by six is one and taking out a why from why squared will leave me just with one. Why by itself, right, Because we have a wide of the 1st 1 of the first. Those being multiplied together would give us why squared time. So now I have one x y coming from this initial term, and now we need to take out six wife from this second term. That's easy. We just take out the coefficient and we leave it with the coefficient of one so plus one times the square root of three x Y. Now removing these coefficients of one because they're unnecessary leaves us with six. Why times a quantity of X y, plus Route three X y as our final answer.

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