in-3-41-express-each-product-in-simplest-form-variables-in-the-radicand-with-an-even-index-are-no-32

Question

In $3-41$ , express each product in simplest form. Variables in the radicand with an even index are non-negative.
$$
(x-\sqrt[4]{3 y})(2 x-\sqrt[4]{3 y})
$$

M S · Accepted Answer

So we have X minus or Route three. Why? Tux to X minus four, Route three What? Someone of all the same process of boiling. So you multiply the 1st 2 We have two expired the outer terms. So have minus X times four through. Uh, do you want the inner terms? So we have two x times of 43 there you are. And they&#x27;re gonna multiply the lasting terms they will have before through the three. Why times four critically. Why? Since it&#x27;s the same thing conduced square, that So now we will do simplified further. So because these have the same base, we can combine the coefficient. So this is the negative politics, and this is a negative to it We&#x27;re gonna bring down is to x squared and the negative 11 is to give us negative three x So I support through a 31 And with this last term here, this square, usually it cancels out with our radical. But because we have a four here, we divide or bar too. So just become spirited three wives. So in case it, you need a further explanation, let&#x27;s plug in a simpler example. So let&#x27;s say the 432 16 squared. This is really the fourth through of 16 times of 43 to 16. Now we know that the board through the 16 is, too, because two times, two times, two times to give us 16. So this is two times two and never give us four. So this shows that the radical does not just cancel out completely, but we just have to divide it by two. But that&#x27;s how this radical is still here, and this is our final answer, as we cannot simplify it for this.

Sarah Wharton · Answer

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&#x27;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&#x27;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&#x27;re going to combine like terms inside this radicals were to do the 12 times three we&#x27;re gonna do X times x we&#x27;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&#x27;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&#x27;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&#x27;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&#x27;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&#x27;s what I&#x27;m gonna do right here. I&#x27;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&#x27;re gonna continue to solve. So it looks like over here I never multiplied the three and two together, So I&#x27;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&#x27;s just the number on the inside. So this part right here. This why is going to get replaced? So I&#x27;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&#x27;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&#x27;m going to combine like terms. First of all, I&#x27;m gonna combine this three x and why which are all in the routes into one big route. And I&#x27;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&#x27;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&#x27;s gonna be the coefficient out in front. I&#x27;m gonna put out a six and a why. And I&#x27;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&#x27;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&#x27;re unnecessary leaves us with six. Why times a quantity of X y, plus Route three X y as our final answer.

M S · Answer

So we have spirit of exit. The fifth time&#x27;s walk you time square, the three ex swat Something&#x27;s going to separate everything or combine everything to be under the same radical. So we have extra defense. Why Q three x and why? So we also know that will we have the same base of different exponents and were multiple ending Together we end up adding those exploited so X to the A tough Next will be will give us extra be a plus feet. Linda the same. Think you&#x27;re under these radicals, so whenever he doesn&#x27;t have a next opponent, hasn&#x27;t obviously exploded. Avoid close to excess first, so have fixed to the five. Let us one and then we do Wise will have block to the three plus one, and they will just have our three. This turns into where of X to the suit. Times wire to the Fort Times three. So whenever we have the square root, we have this understood to hear because it&#x27;s square root and everything always has a lemon said, explaining to avoid so square root. Written as a fractional exponents is really next to the 1/2. So you take the number undo the radical, and you place it over the number outside of the radical. What about those girls? Always haven&#x27;t understood to here. So if we separate, this will have square root, uh, exit six times square root. Uh, want to the Fort Times the square root last 30. So we change everything to have fractional exponents Will have x to this six over too. Times log onto the floor over to Times Square three. You don&#x27;t have any reason to change that. Well, I just want to try. And she used to have actual next one. It&#x27;s because we can reduce these fractions. So six divided by two will give us three. So it&#x27;s a little bird. What about? About to give this to so r squared and we&#x27;ll still have this square root three just sitting here. Well, this is our final answer.

Sarah Wharton · Answer

we are solving for the square root of five y times the square root of four y cube. We combine this into one big radical sign, the square root of five. Why times for why I cubed five? Why is the same thing is five y to the first power, which allows us to rewrite this as the square root off five times four times Why to the one plus three? Because when we multiply variables together like why the one and y cubed? We add their exponents. This leaves us with the square root of five times, four times y to the fourth. You&#x27;ll notice I didn&#x27;t combine the five and four together because in our next step, we need to break them apart. Anyway. This is equal to the square root of five times the square root of four times the square root of Y to the fourth, and our next step. We&#x27;re going to simplify each part. Individually. The square root of five stays the same. The square root of four becomes, too, in our square root of why the fourth becomes why squared? Our final answer has to be in a specific order coefficient, then variable than radical. So it&#x27;s equal to two. Why squared times the square root of five? And that&#x27;s our final answer.

Sarah Wharton · Answer

for this problem, we&#x27;re solving three Route three squared, which can be expanded as three route three times three, Route three. To solve this, we multiplied together the whole numbers in front and we multiply the numbers inside the radical signs. So this will be equal to three times three times route three times Route three. This part, the three times three is equal to nine. And the second part here could be combined in the same radical as the square root of three times three. Bringing this down to another line. This is equal to nine times the square root of nine. We know that this word of nine is equal three. So this is equal to nine times three, which is equal to 27 that is our final answer.

M S · Answer

So we have the port through of 27 time support, 3 to 3. So because the radicals are just saying, we can combine everything I need to radical so you can have 27 tons. Three. Only the radicals 27 times three Is this aging one and the fourth word of anyone fourth route, where it&#x27;s just, like swear route. But it&#x27;s four numbers board that four of the same numbers multiply together. And now to give us a tingly in this case, the fourth or debating Willingness. Three. That&#x27;s three times. Three times, three times three will give us a tea boy. So the fourth we&#x27;re debating witness three.

Problem

In $3-41$ , express each product in simplest form…

Problem 34 Hard Difficulty

In $3-41$ , express each product in simplest form. Variables in the radicand with an even index are non-negative.
$$
(x-\sqrt[4]{3 y})(2 x-\sqrt[4]{3 y})
$$

Answer

$2 x^{2}-3 x \sqrt[4]{3 y}+\sqrt[2]{3 y}$

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Ann Xavier Gantert

Algebra 2 and Trigonometry

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$2 x^{2}-3 x \sqrt[4]{3 y}+\sqrt[2]{3 y}$

Algebra 2 and Trigonometry

Ann Xavier GantertAlgebra 2 and Trigonometry

Ann Xavier Gantert

Algebra 2 and Trigonometry