in-3-38-write-each-expression-in-simplest-form-variables-in-the-radicand-with-an-even-index-are-n-22

Question

In $3-38$ write each expression in simplest form. Variables in the radicand with an even index are non-negative. Variables occurring in the denominator of a fraction are non-zero.
$$
\sqrt{162 a^{4} b^{3}}+3-a b \sqrt{18 a^{2} b}-1
$$

Finian Lickona · Accepted Answer

So in this exercise, we&#x27;re gonna be simplifying the expression radical. 162 8 of the fourth beat of the third plus three minus a be radical 18 a squared B minus one. Okay, so the first thing I&#x27;m gonna do is try to simplify this radical as much as I can, and it&#x27;s probably gonna match up with this radical in some way. So the first thing I recognize is I can rewrite this as two times 81 81 is a perfect square. I can factor that out. Can we write it in the fourth as a squared times a squared and I can rewrite beat of the third as b squared. That was a bad squared B squared times beat. So if I factor all these perfect square terms out, I&#x27;m gonna factor this 81 out and get a nine. I&#x27;m gonna fact this a squared out in this other a scored out. So that&#x27;s eight times a So it&#x27;s another a squared. I&#x27;m gonna factor this b squared ounce that&#x27;s gonna be be out front in here. And I am left with a two in the radical and A B in their article. Okay, so let&#x27;s see if we can do the same thing to this radical. Obviously, 18 can be re written as two times 99 is a perfect square. A squared is already a perfect square, so I can factor that out in front as well. So this radical becomes a B times three times another A. And I have a to be left in the Radicals. This whole expression is going to be three a squared B radical to be. And so and then obviously these unit terms right here can be combined separately. So three minus one is going to be too. So we&#x27;re just gonna put that over there. For now. We have nine a squared B minus three a squared, be on the right, audible to bees. That&#x27;s gonna be nine minus three. So that simplifies to six a squared B square root to be and then we still have a plus. Two on the outsides are final. Answer is going to be six a squared, be radical to be plus two

Julie Silva · Answer

in this problem. We&#x27;re being asked to simplify the given expression. Well, we&#x27;re adding radicals, so the first thing we need to do is simplify both of our radicals. So let&#x27;s start with our first one. Well, we have the square root of 24 B to the third. What we can square. We&#x27;re sorry. Simplified the square of the 24 by rewriting it as the square of the four attempt to square the six since four is a perfect square. So I&#x27;m gonna bring down the four B that&#x27;s in front. And then, like I said, I&#x27;m gonna rewrite the square of the 24 as to square the four Times square the six Then we have beaten the third will remember. It would only be a perfect square if the US bone it was even. So I&#x27;m gonna rewrite that as b squared. Our sorry, the square root of B squared times the square of the B. Since B squared times beat is be to the third. All right, let&#x27;s go to our second radical. We have the square of the 54 B to the fifth. Well, I can rewrite the square with the 54 as to square with a nine time six since nine is a perfect square. And as I mentioned before, we want to rewrite beat to the fifth, so it hasn&#x27;t even power in it. So I&#x27;m gonna rewrite that as the square root of B to the fourth times to square the B sense be to the fourth times B is in fact, be to the fifth. All right, let&#x27;s go ahead and simplify. So from our first term, I&#x27;m gonna bring down the four B. Then we have to square with the four which is to and square with a B square. This be now We can&#x27;t simplify the square of the six or to square with a B, so I&#x27;m gonna leave that as a square root of six B. Now let&#x27;s go to our second term. Well, we have a square with a nine, which is three. We have the square with the B to the fourth, which is B squared. And again we can&#x27;t simplify the square of the six of the square the be So we leave that as the square root of six B. Now take a look at that first term. Let&#x27;s multiply our coefficients. We have four B times to be, which is eight b squared. So we&#x27;re left with eight B squared times the square root of six B, and we&#x27;ll bring down the second term. Three b squared times the square of the six b. Okay, perfect. Now both of our radicals have been simplified, and these two terms are in fact like terms because both radic cans of the same. So we just need to combine the coefficients well. Eight. B squared plus three B squared is 11 b squared, so we&#x27;ll have 11 b squared. And then we&#x27;ll bring down our radical to square with the 60 and now was simplified completely. So our final answer is 11 b squared times the square root of six B.

Finian Lickona · Answer

So in this exercise, we&#x27;re going to be simplifying the expression eight Radical 11 Be toothy fourth minus radical 99 Beat the fourth. Now our first step is we want to transform this radical until looking something like this radical. So we recognize we can rewrite this as nine times 11 b to the fourth. Just nine times 11 and nine is a perfect square, so we could factor that outside of the radical. So we have eight 11 beat of the fourth minus three 11 beat fourth. Now this is almost done, except that we haven&#x27;t quite simplified both of these radicals because beat of the fourth can be rewritten as b squared times B squared so we can actually factor b squared out of the radical so we can rewrite. This is eight b squared radical 11 minus three B squared radical 11 and we can group thes coefficients together and simply write this as eight minus three b squared radical 11. Which is, of course, just five B squared radical 11

Finian Lickona · Answer

in this exercise, we&#x27;re going to simplify the expressions for a B squared minus a B squared. Okay, so it&#x27;s easy to recognize immediately. Four is a perfect square. B squared is a perfect square, so we can fact those both out of the radical as to be in there. We are left with a inside the radical. Ah, And then, of course, this b squared can also be factored out. So with this well, it wouldn&#x27;t be radical. A So now we simply have to be radical. A minus be radical. A. So that means our final answer is simply be radical.

Finian Lickona · Answer

all right, so in this exercise, we are going to be simplifying the expression 18 b squared, plus a radical 18 B squared plus radical 800 b squared. It&#x27;s the first thing we want to do is find ways to break this down into a product of some number in a perfect square. So 18 is nine times to nine is a perfect square. We can factor it. Outside of the radical is three we&#x27;re left with two and then b squared obviously is a perfect square sunken factor that outside of the radicals were left with three be radical too. Now 800 can be factored as two times 404 100 is a perfect square. It&#x27;s 20 squared, obviously be squared is still a perfect square. So we&#x27;re gonna rewrite that as 20 be radical tube. Now we can combine our coefficients on our radicals So three b plus 20 be simply 23 b radical too

Julie Silva · Answer

we&#x27;re being asked to simplify to given expression. Remember, when you subtract radicals, you need to simplify both radicals first. Well, let&#x27;s take a look at the square root of eight A. Well, the perfect square four goes into 82 times. So I&#x27;m going to rewrite this as the square of the four times the square root of two way because four times to a is a day now I can&#x27;t simplify the square of the two way, so I&#x27;m gonna lead. That is this. Now let&#x27;s go back to our first term. We have the square with a four witches too. And we can&#x27;t simplify the square of the two way. So we bring that down. So we have two minus the square that you weigh minus the square that you weigh. All right, so now both radicals air simplified and we can combine these because both radic cans of the same. So we just combine the coefficients which remembered the coefficient of our second term is one. Well, two minus one is just one. So we&#x27;re just left with the square root of two a

Problem

In $3-38$ write each expression in simplest form.…

Problem 24 Hard Difficulty

In $3-38$ write each expression in simplest form. Variables in the radicand with an even index are non-negative. Variables occurring in the denominator of a fraction are non-zero.
$$
\sqrt{162 a^{4} b^{3}}+3-a b \sqrt{18 a^{2} b}-1
$$

Answer

$6 a^{2} b \sqrt{2 b}+2$

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$6 a^{2} b \sqrt{2 b}+2$

Algebra 2 and Trigonometry

Ann Xavier GantertAlgebra 2 and Trigonometry

Ann Xavier Gantert

Algebra 2 and Trigonometry