College Algebra: Radical Notation, Root Properties

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We've taken square roots and we think about these things and that's all well and good but I really want us to take a real close look at square roots and radicals and what they mean and how we can think about it in different ways. Let's just start off with a simple thing to get us going. If I say square root of 16, what is that? Well, that equals four and why? Because 4(4) = 16. No big mystery there. The square root means finding the value so that when I take that value and square it I get back to here. This is undoing 4 squared = 16 basically.
If I wanted to write that square root as a power, what power would it be? Well, let's think about that for a second. What I want to say is 16 to some mystery power, I don't know what that power's going to be but I know that whatever that power is it's going to equal four. My challenge right now is to figure out what is the power analog of square root. Let's think about. 16 we know is 4^2, so I could write this as 4^2. That's just 16. That's the 16 part. Don't forget that I'm raising it to this mysterious power and that's supposed to equal 4.
And what about laws of exponents? We talked about this that if I have something raised to a power, I'm raising all that to a power. The net result is that I multiply these two things so I'd have 4^2 times question mark power = 4. If I see 4 to some power = 4 to some power, well those powers must be the same. What's the power on 4 here? There's an invisible one power, 4^1 right here and since these bases are the same, we must have those powers being the same which means that 2 times question mark has to equal 1 or, in other words, that mystery exponent turns out to be , and that's sort of peculiar that the power I have to raise 16 to in order to get 4 turns out to be the power, but that's what it is. It turns out that square roots are the same thing as raising something to the power. Now to answer our question, what I see is 16 to the power is the same thing as the . They both equal 4 and in general, let me just say that if you take the nth root of some number you can write that as that number to the 1 over n power. If you want a cube root you could think of that as something raised to the 1/3 power and the reasoning that we just went through will always work.
Let's do some examples together. For example, if I say the fifth root of 32. What number is that? Well, I've got to think of a number so that when I multiply it by itself five times I get 32, and if you think about it, 2 is the only number that really works. If I take 2 and multiply it by itself, 2(2) = 4(2) = 8(2) = 16(2) = 32. That's it. I could think of this as 32^1/5. Either way is fine.
Let's try some other radical things where I start to use the power of this expression. For example, suppose I take the cubed root of 7^15. That seems threatening and complicated but the thing to do is not panic but instead think about this conversion because I could actually write this, if I wanted to, as 7^15 and then all raised to what power? Well since it's a cube root I'm raising it to the 1/3 power and what do I do with exponents now? I always multiply the exponents, so this would be 7^15/3 which is just 7^5. In fact, the cubed root of 7^15 is nothing more than 7^5. You can actually simplify very radical expressions by using the power of this fact right here.
Let's try some more. Let's try one where it gets a little bit exciting. How about the 7th root of -10^7? That seems sort of threatening. Let's think about that for a second. Well, on the one hand we might be able to figure this out and reason together. What is the number so that when I raise it to the seventh power I get this thing in here? Well, if you look at that you'd see it's just -10 because -10 raised to the seventh power is that. In fact, this is the 7th root of -10^7. That's always true. If you take the root of something raised to the same power as the root you're taking, you'll always get the number when this thing here is an odd.
Let me show you a very annoying example, . Let's work this out by hand. -5^2 is -5(-5); negative times a negative is a positive. This is 25. = 5, so notice the difference here. It's +5 and that's because I'm taking an even root. Even roots like this are positive. You've got to be really careful. If you're taking an odd root and you have a negative then you're going to have a negative answer but if you're taking an even root like this, the answer will be positive. There's a difference between taking odd roots and even roots and the reason comes back to the fundamental fact about multiplying negatives by itself. A negative times a negative is a positive. Every time I have two negatives together they produce a positive but if I've got three negatives together then I've got negative, negative, negative. These two negatives make a positive but then when I tack on this one more negative I get a negative. Odd numbers of negative signs produce a negative. Even numbers of negative signs produce a positive and that's why there's a distinction between taking odd roots versus even roots. That's a really, really important thing to be leery of.
You can also take roots of all sorts of crazy looking things. Let me show you that, in fact, you can use all the properties that you've already learned. Square root of this whole fraction, well you could think of this as the square root of the top divided by the square root of the bottom and if you take the square root of the top you get 11. If you take the square root of the bottom you get 7 and you can always simplify radicals by just using the basic properties that we already know about fractions and how these things work if you're raising things to powers or taking roots.
Let me show you one last thing. Let's suppose you take any number, I'll call it a for any number but it's not zero. If it's not zero, then I can certainly divide by it and what is any number divided by any number? As long as it's not zero I could cancel them away and I get 1. On the other hand, let's use the properties of powers. The power here is a 1. The power here is a 1. And what's the law? The law is you subtract the exponents and so I see 1 -1, which is zero, and I've discovered a fantastic fact. If you take a number that's not zero and raise it to the zero power, it has to equal one because it cancels out and so there's a great fact that will come in really handy that if you take a number and raise it to the zero power, it equals one, any number as long as it's not zero because then you can't divide by zero. For example, if someone comes along and says to you, "Can you calculate this, 7.3981^0?" You say, "Yes, I can." And you say, "It's 1." Because it's just 7.3981 divided by 7.3981. They cancel and you get one. That's a great fact about exponents. You can actually think about radicals as radicals or you can think about them as fractional exponent. Either way it's still radical.
I'll see you at the next lecture.
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Radical Notation and Properties of Roots Page [2 of 2]