Product & Quotient Rules And Simplifying Radicals, Part 3 5 - 3c

This video is going to be about the quotient rule for radicals, and this is kind of similar to the product rule. What the quotient rule says is this. If we have a root, let's say a square root, and we have a fraction underneath, so we have the square root of a over b, we can break that down into two fractions, each one with a radical. So in other words, the square root of a over b equals the square root of a over the square root of b. And of course we can go in the other direction as well. So let's look at some places where you could use it. Let's say you've got the problem the square root of 36 over 25. Well, if you go ahead and divide 36 by 25, you're gonna get an awful number, and you don't want to take the square root of it. However, if we break this down into the square root of 36 over the square root of 25, it becomes a whole lot easier. So 36 is a perfect square... the square root of 36 is 6. 25 is a perfect square. The square root of 25 is 5. So, we started out with this big fraction, broke it down into two radicals, and we end up with a simple fraction. Here's another example. I've got the square root of 32 over 2. Now the reason I put this in is this... just because you could break this down into two square roots, it doesn't mean you want to. Before you do anything, look at what that fraction is and ask yourself what would happen if I divided 32 by 2. Well, if you divide 32 by 2, you get 16, and 16 is a perfect square, 16 is 4 times 4 . So in other words, uses this rule when it makes sense, but don't just use it just because you can. Now, as I said, you can take this rule and use it in the other direction as well, you can start out with two radicals and put them both under one radical sign. So here's some examples of that. Here I've got the square root of 18 over the square root of 2. Neither of these is a perfect square, but I know that if I divide 18 by 2 I'll get a 9. So that's gonna be nice. I'm gonna take this, make it into one radical, the square root of 18 over 2. That's the same as a square root of 9, and the square root of 9 is 3. Once again, just because you can do something, it doesn't mean you necessarily want to. I don't want to make this into the square root of 3 over 4. 4 is a perfect square, so let's just take this and make it into the square root of 3 over 2. I can't do anything with the square root of 3, I can't simplify it, so this is my answer. As we saw with the product rule, this works for higher level roots as well. So this works for third roots, or cube roots. So here I've got the third root of 16 over the third root of 2. Neither of these is a perfect cube, but if I divide 16 by 2 I'll get an 8. So let's do this in two steps. I've got the third root of 16 over 2. 2 into 16 is 8. So I've got the third root of 8, and because you've memorized your list of perfect cubes, you know that 8 is a perfect cube, and this whole thing just becomes 2. Okay, more examples. This just a slightly more complicated one because we've got something multiplying the whole fraction. but before we can deal with this 3, let's look at what's under the radical sign. I've got the square root of 49 over 81. Now I know that both of these are perfect squares. 49 is a perfect square and 81 is a perfect square, so it's going to make sense to break this down into two radicals. That means I'm going to have 3 times the square root of 49 and that's going to be over the square root of 81. So taking the square roots, I'll have 3 times... the square root of 49 is 7, the square root of 81 is 9, and just looking at these... before I multiply 3 times 7, I realize that I can factory a 3 out of the numerator and the denominator. So it makes more sense to do as my next step. So dividing the 3 by 3 I get 1. Dividing the 9 by 3 I get 3. And that means my answer is just going to be 7/3. Okay. And this also works for variables. So let's do a problem with some variables in it. Here I've got 2 times the square root of 50 a to the third, b to the fifth, c to the fourth all of that over the square root of 2 a b c squared. Okay, now, I could pull a c out... I have this c squared, so I could do something with that, but looking at the other numbers, dividing 50 by 2 would really be nice, because that would give me 25, and if I had a to the third over a, I could reduce that, and b to the fifth over b could be reduced. So let's make this all into one radical. So I've got 2 times 50 a to the third b to the fifth c to the fourth over 2 a b c squared. And now we'll reduce this fraction. So I'm going to have 2 times... 50 divided by 2 is 25, a to the third over a... I'm subtracting this exponent, the 1, from the 3, so 3 minus 1 is 2, that would be a squared. b to the fifth over b... so I'm subtracting 1 from 5... will give me b to the fourth. And c to the fourth over c squared... subtracting the 2 from the 4... I'll get a c squared. Now this should be really easy to simplify. So I'm going to have 2 times the square root of 25... the square root of 25 is 5. the square root of a squared... dividing the 2 index into this 2, I just get a 1. So that's an a. b to the fourth... dividing 2 into 4 I get a 2... so I'm going to get b squared, and dividing... taking the c squared and dividing 2 into that exponent 2, I'll get a c, and now all I have to do is multiply my 2 times 5. So that will give me 10 a b squared c. And that's going to be my simplified version of the original problem. So remember, the basic rule says that you can take two radicals which are present as fractions and put them under one big radical side with the fraction under the radical sign, or you can break this up and go the other way. All the other rules you covered are still going to apply. So you're going to do your square roots and third roots and so on, just as you did before. Okay, get plenty of practice for these, they're really simple once you get the hang of it, and it's kind of nice to see something as complicated looking as this turn into something as simple as that. Okay, take care, see you next time.