in-3-38-write-each-radical-in-simplest-radical-form-variables-in-the-radicand-of-an-even-index-ar-36

Question

In $3-38$ , write each radical in simplest radical form. Variables in the radicand of an even index are non-negative. Variables occurring in the denominator of a fraction are non-zero.
$$
\sqrt[4]{\frac{2 a^{8}}{b^{2} c^{3}}}
$$

James Kiss · Accepted Answer

if we have the expression the fourth root of to A to the eighth over B squared C to the third. We want to simplify this well, according to some properties of radicals A. When we&#x27;re rooting a fraction, we can take the in this case, fourth root of the numerator over the fourth root of the denominator. No, we can take the fourth root of aid to the eighth. That is a squared, so we have a square times before through of two over the fourth root of B squared C Cube. Now we need to rationalize the denominator, so we need to scale both the numerator and denominator by a radical that&#x27;s going to remove it from the denominator so we can only multiply four through its by fourth roots. And if if fourth roots are going to a value, excuse me, evaluate powers of four that we have to think about, what could we multiply? What&#x27;s underneath the radical to get powers of four. So if you already have a B squared, if we do, another B squared will have be to the fourth, which then we can evaluate the B and if we already have C to the third. We just need to multiply by another C to get seated the fourth. Therefore we could multiply it. So that&#x27;s what we&#x27;re going to scale in order to rationalize the denominator. Yeah, And so now what we have here in the numerator, we still have that a squared. And then we can multiply fourth routes together, we&#x27;ll have times the fourth root of two B squared C over. Now, this is the fourth through a B to the fourth, which is B and the fourth root of sea to the fourth, which is C.

Finian Lickona · Answer

All right, so we want to write the term eight Ah, b Cubed square root in its most simplified radical for So we&#x27;re gonna look for ways in which we can break each of these terms down into a product of some value and a perfect square. So if we look at the number eight, we recognize that we can write it as two times four. And if we look at the quantity, be cute. We recognize we can write it as b squared times be now, the square root of four is too. So we could take that out of the radical. Put it out here and this square root of B squared is be so we can also take that out of the radical and put it right here. So our final answer is going to be to be hang on to be. And this too has to stay in the radical because it can&#x27;t be simplified. And this be also has to stay in their radical. So our final answer is to be rude quantity to be

Nick Johnson · Answer

we have a square root of for a cube over 27. Be cute so well, we can rewrite this as two square root off for a cube over the square root of 27. Be cube okay, And then, while the square root of for a cube that&#x27;s the square to four times the square root of a cube, the square to four is a perfect square that&#x27;s equal to this is equal to two times, while the square root of a cube over what, a square 27 B cube that&#x27;s equal to the square. Root off 27 is nine times three. Okay, times be cube, but it&#x27;s got a nine history, so this is equal to two square root of a cube over square. Tonight is threes that can come out of outside the radical. We&#x27;ve got three, and we still have a three left. Inside the radicals, we have three times the square root off three b que. I guess we could go one step further here and say, Well, a cube is really a square times A. But in the square root of a square is a so we have. This could be equal to two times the square root of eight. Some excess because good of a cube is well, It&#x27;s the square root of a squared times the square root of a but the square a square is a salt lips. So when a should come out here and age can come out, you have to a times the square root of a okay over well, three times the square root of three b cube. But again, be cube is B squared times B. We can pull out the square root of B square, which is be so what? We have over three B and then under the radical, we still have a three beat. So we have to a squirt of B over three b times the square root of three. All right, take care.

Finian Lickona · Answer

So for a fix this exercise, we want to simplify the term 48 No, A to the ninth B to the third. And this is the fourth through. So we need to look for ways in which we can decompose each of these terms into fourth power route. So we think about our fourth powers there one, um gosh 16. And we already see that 48 is the product of 16 3 So we can rewrite that as the fourth root of so, uh, 16 and three. And then if we look at this age of the ninth term, we see that we can rewrite that as eight of the fourth times eight of the fourth time&#x27;s A and we can to the fourth route. And then we can write the beat of the third term just as beating the third so we can pull this 16 out of the fourth radical and that becomes a two. So this is a two route three. We can pull this aid of the fourth out and fronts that becomes an A and we could do the same thing that this a So it&#x27;s in eight times in a So we get a squared and we have to leave that last A still under the fourth through radical and the beat of the 3rd 4th through term cannot be simplified anymore, so we&#x27;re just gonna have to leave that the way it is. So our final answer is going to be to a squared times three a beat of the third 43

Finian Lickona · Answer

So for this exercise, we&#x27;re going to be simplifying the term route 15 over eight. Be cube. Now 15 cannot be decomposed into any perfect square, so it&#x27;s gonna have to stay inside the radical. However, the one over eight term can be rewritten, as is gonna include that 15 right there. One over two times four and four is a perfect square, so it can be taken outside of the radical. And then we can rewrite that beat of the third term as one over B squared times one over B. And of course, this one over B squared turn can be taken outside of the radical. So if we take that four outside of the radical becomes a 1/2. And if we take this B squared term outside of becomes one over B So our final answer is going to be 1/2 be radical. 15 over to be

Finian Lickona · Answer

So for this exercise, we are going to be simplifying the term square root of three beat of the third over 44. Now the same principle of radical simplification applies even when you have a fraction, you just have to make sure to keep terms that are in the denominator in the denominator outside of the radical. So we can rewrite this as one over four times 11 times three B squared times be so we can bring this one over four out in front of the radical as 1/2 the one over 11 remains and we can bring this B squared term out in front, as be so. We have won over 11. We leave the three in there and we leave the single be turbine there. So our final answer is 1/2 be Route three be over 11.

Problem

In $3-38$ , write each radical in simplest radica…

Problem 37 Easy Difficulty

In $3-38$ , write each radical in simplest radical form. Variables in the radicand of an even index are non-negative. Variables occurring in the denominator of a fraction are non-zero.
$$
\sqrt[4]{\frac{2 a^{8}}{b^{2} c^{3}}}
$$

Answer

$\frac{a^{2} \sqrt[4]{2 b^{2} c}}{b c}$

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$\frac{a^{2} \sqrt[4]{2 b^{2} c}}{b c}$

Algebra 2 and Trigonometry

Ann Xavier GantertAlgebra 2 and Trigonometry

Ann Xavier Gantert

Algebra 2 and Trigonometry