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Evaluating Radicals

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Hello, and welcome. My name is Anthony Varela. And this tutorial is about evaluating radicals. So what we're going to do is evaluate some square roots. We'll look at how to evaluate cube roots. And then we'll take a stab at evaluating higher order roots.

So, first, let's review radicals and exponents as inverse operations. So if I have 4 and I cube 4 and then I take the cubed root of 4 cubed, that brings me back to 4. Cubing and taking the cubed root are inverse operations. So taking the nth root of an expression to the nth power undoes the exponent.

And I can take 4 and cube root 4, and then take that whole quantity and cube that, that will also bring me back to 4. So I can do it in the other direction, too. Raising an nth root to the nth power undoes the radical. So with our radical expressions, the nth root of a, the inverse operations are the root and the power. And we should note that this only works with non-negative values of a here.

So now let's then talk about how to evaluate square roots. And a piece of advice is look for perfect squares. This makes it easy to do in your head mentally without a calculator. So, for example, the square root of 25, I know that 25 is a perfect square. Now, what does that mean?

Square roots of perfect squares are integers. So what I can think of is 25 is 5 times 5. So the square root of 25 is 5.

How about the square root then 64? Well, 64 is also a perfect square. And when we're talking about perfect squares, the square roots of those are integers. So 8 times 8 equals 64. So the square root of 64 is 8.

Now, if you don't have a perfect square underneath a square root, you can always use your calculator to evaluate that radical. So, for example, here, we have the square root of 15. 15 is not a perfect square. So this isn't going to be a clean integer.

So, using your calculator, there a couple of different things you could do. Your calculator might have the radical science. So this is a Square Root button. And then you'll type in 15. And then you'll type in Equals, and it will give you that decimal approximation.

Other ways is you can type in your Radicand 15 and use a fractional exponent. So type in your Exponent button. And type in 1/2 grouped in parentheses so your calculator knows exactly what to do. That will work, as well.

So now let's talk about evaluating cube roots. And just like with square roots, the hint here is look for perfect cubes. So, for example, the cubed root of 27 is going to evaluate to a nice clean integer because 27 is a perfect cube.

What does that mean? That means that there is a number I can use in a chain of multiplication three times to get 27. So cubed roots of perfect cubes are integers. 27 equals 3 times 3 times 3. So the cubed root of 27 then equals 3.

Now, let's take a look at this one. The cubed root of negative 64, and this is actually the first time we've seen a negative number here. Well, believe it or not, negative 64 is a perfect cube because negative 4 times negative 4 times negative 4 equals negative 64. So this is different from our square root example.

A negative number multiplied together three times will give us a negative number. That actually holds true for any odd root. This could be a fifth root, or seventh root, or a ninth root and you're multiplying a negative number odd number of times, you get a negative number.

But when you multiply a negative number together an even number of times, you always get a positive. So when you have an even root of a negative number, that doesn't actually exist as a real solution. But you certainly can have a negative underneath a radical with an odd root and get a real number solution.

So this cubed root of negative 64 equals negative 4. All right. And how do we do this using a calculator if we wanted to evaluate a cubed root using your calculator? So here we have the cubed root of 44. 44 is not a perfect cube. So it won't be a clean integer.

Using your calculator, you could type in a Cubed Root button, if your calculator has it. And then just type in 44 and Equals. And it will give you that decimal approximation.

Or you can type in 44. Your calculator might have this button, which says you want to perform a radical. And then you'll just have to type an index next, which would be 3, because we're talking about a cubed root.

Or what you could do is use a fractional exponent again. You can have 44, your radicand, the expression underneath a radical. Raise that to a power. And this would be 1/3 this time because you're dealing with a cubed root. That will also get you an answer on your calculator.

All right. Well, how do we evaluate a higher order roots? We certainly can do this on your calculator. And I'll show you how to do that in a minute. But one strategy is to take your expression underneath the radical and do what's called "prime factorization," which is actually breaking down that number into its prime factors.

So I'm going to take 243. And I'm going to divide it by some prime numbers as much as I can. Well, I know that it's an odd number. So it won't divide by 2. So I'll divide by 3. 243 is 3 times 81. Now three is prime, but 81 isn't. So I'm going break down 81 some more.

81 is 3 times 27. Well, 27 is 3 times 9. And 9 is 3 times 3. So now what I have are all the prime factors of 243. It's 3 times 3 times 3 times 3 times 3. So I can rewrite this fifth root then as the fifth root of 3 times 3 times 3 times 3 times 3 since I know this equals 243.

Well, the cool thing here is that if you see a factor five times, because we have the fifth roots, it just evaluates to that factor. So I see the number three here 1, 2, 3, 4, 5 times. So the fifth root of this just simply equals 3. And that's pretty nice.

But oftentimes it's not always that nice. So how do we do this using a calculator? So let's evaluate the seventh root of 145. Sounds like a complete mess. But using your calculator, this is what you do.

You type in your radicand, 145. You might have that Radical button on your calculator. And then you'll just need to type in the index, which is 7. So that can get you your answer.

Or you can, for any nth root, you can type in your radicand, so that expression underneath the radicals, so it would be 145. And do a fractional exponent. So we're raising it to a power of 1 over n, whatever that index is. So, in this case, r would be 145 and n would be 7. That's the way that you can evaluate any nth root using your calculator.

So what did we talk about today? Well, we talked about a radical expression. This is actually to say the nth root of a since we can take more than just the square root of a number. The inverse operations here are roots and powers. And so this only works for positive or non-negative a values and our n's have to match. So that would be the fourth root and the fourth power, eighth root and the eighth power, et cetera.

We talked about perfect squares. The square roots of perfect squares are integers. And perfect cubes, the cube roots of perfect cubes are integers. We also talked about prime factorization as a way to evaluate higher order roots. So we're looking for a factor n number of times evaluates to that factor when we're talking about nth roots.

And using your calculator, you can evaluate any radical by raising it to a fractional exponent. For the nth power, that fraction would be 1 over n.

Well, thanks for watching this tutorial on evaluating radicals. Hope to see you next time.