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Evaluating Radicals

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SPEAKER: Today we're going to talk about evaluating radicals. Remember a radical is something like a square root or a cubed root. So we're going to briefly review how radicals and exponents are inverse operations of each other. And then we'll do some simple examples.

So exponents and radicals are inverse operations of each other. Let me show you an example to further demonstrate this. So let's say I had three to the second power. Three to the second power means that I'm multiplying three by itself two times. And so three times three is going to give me nine. Now if I use a radical-- if I take the square root of nine-- taking the square root means what number do I multiply it by itself two times to give me nine. And that number is three. So we started with three and we squared it, bringing us to nine. If I take the square root of nine, I'm going to get my answer back at three. So in this way exponents and radicals are inverse operations of each other. Let's do some examples of how to simplify radicals.

So let's do some examples of evaluating radicals. All right. So my first example is the square root of 25. Whenever you're taking the square root of a number you want to start by thinking, is this a perfect square. Because then you wouldn't have to use your calculator. So 25 is a perfect square because I know that five times five is 25. So the square root of 25 is going to give me five. Remember perfect square is always going to evaluate to be an integer. Similarly for the square root of 100. 100 is a perfect square because I know that 10 times 10 is 100. So that means that the square root of 100 is just going to give me 10.

Now the square root of negative 16 is actually not a real answer. It will not evaluate to be a real number because even though positive 16 is a perfect square-- four times four will give me 16-- I cannot take the square root of a negative number, because there's no number that I can multiply twice and get a negative answer. And that's because, remember when you multiply two negative numbers you're going to get a positive. And if you multiplied a positive and a negative together you would also get a negative. And those two numbers would not be the same. So you cannot take the square root of a negative number. So this will give me not a real number.

All right. So then we've got the square root of 43. Now 43 is not a perfect square. So I'm going to have to use my calculator to evaluate that radical. So I punch in the square root of 43 and I'm going to get approximately 6.557. So 43 is not a perfect square. It does not evaluate to be an integer.

All right. Let's look at some cubed root examples. So 1,000. Again, we're going to start by thinking if it's a perfect cube. And 1,000 is, because 10 times 10 times 10 is going to give me 1,000. So the cubed root of 1,000 is 10.

Now we've got the cubed root of negative 125. Now this is different. Taking the cubed root of a negative number is different from taking the square root of a negative number, because there is a number that I can multiply by itself three times and get negative 125. A negative times a negative times a negative is going to give you a negative answer. So negative 125 is actually a perfect cubed also, because negative five multiplied by itself three times is negative 125. So the cubed root of negative 125 is negative five.

The cubed root of 204. Now 204 is not a perfect cube. So again we'll have to use our calculator to evaluate that. And that will give us approximately 5.887.

So these are examples of taking the square root and the cubed root of numbers. But we can also evaluate a radical for any number for any root. So you can have any number here in the index of the radical. So the fifth root of 7,776 is going to give us 6. And I know that that's going to be an integer, because if I were to break down 7,776 into a chain of multiplication that's going to give me five 6's multiplied together. So since I'm multiplying five times, and the index of my radical is also five, this is going to give me an integer answer of six.

Now one last key point about evaluating radicals. When you're using your calculator most calculators have a square root sign and a lot of them even have a cubed root sign. But if you are evaluating a radical with the index other than two or three, then there's another way that you can enter it into your calculator. And that's using a fractional exponent. Because remember that you could write a radical. So let's see. Let's say we have the eighth root of 25. We can write that as 25 to the one over eight power. So any time that you want to evaluate a radical and you don't have the radical symbol as a button on your calculator you can also write it as a fractional exponent where the denominator of the fraction is equal to the index of the radical.

So let's go over our key points from today. As usual, make sure you write these in your notes, if you don't have them already, so you can refer to them later. If you use a calculator to evaluate the radical, you first type in the radical button and then you type in the number. If your calculator does not have the necessary nth root button, you're going to use a fractional exponent and type it in as caret, open parentheses, one divided by n, close parentheses. And again, your n number is just the index of your radical. And finally, the nth root of a negative number will not evaluate to be a real number if n is even-- like square root-- but it will evaluate to be a real number if n is odd-- like the cubed root.

So I hope that these key points and examples helped you understand a little bit more about evaluating radicals. Keep using these notes and keep on practicing, and soon you'll be a pro. Thanks for watching.