Hi. This is Anthony Varela. And today, I'm going to introduce radicals. So we're going to talk about radicals and relate them to exponents. Then we're going to talk about how to read and write expressions with radicals. And we'll talk about some common radicals to remember that will help evaluate and simplify radicals much easier.
So to introduce radicals, I'm actually going to first briefly review exponents. So remember that with exponents, we have b raised to the power of n. And we use b in a chain of multiplication n times. So for example, 2 to the 4th power means we're using that base number 2 in a chain of multiplication 4 times.
Well, what does this have to do then with radicals? So here is a radical. This is the square root of 16. And what we're looking for here is a number that we can use in a chain of multiplication 2 times to get 16. So I know that 4 times 4 equals 16. So I could say the square root of 16 then is 4.
Let's take a look at this radical. We can interpret this as we want to know what number we can use in a chain of multiplication 3 times that will give us 8. So I know that 2 times 2 times 2 equals 8 and using this number two in a chain of multiplication 3 times to get 8. So the cube root of 8 is 2. So in general, when we have a radical the n-th power of a, what we're looking for is a number that we can use as a base in a chain of multiplication n number of times to get a. So that's our general radical.
Now, let's relate this then to exponents and specifically that they're inverse operations. So here we have the number 5, and I'm squaring 5. Now, if I take the square root of 5 squared, I'd get back to 5. So the inverse operation of radical is an exponent.
Now, we call this a square root. Now, notice in our previous example, I had a 2 here. And if that's ever blank, we assume that that's an implied 2 or we're talking about square roots. If I take the number 2 and I cube it, I can take the cubed route and get back to 2. And notice here I have to put this 3 there to show that we're talking about a cube root.
Now, we can show this using higher order powers as well and roots. So here we have 3 raised to the 4th power. Taking a 4th root of that, we get back 3. So here we have to write a 4 because we're talking about the 4th root. And for any positive number a, we can say that the n-th root of a to the n equals a. And then notice we're just talking about the n-th root here.
So now let's introduce the language of radicals, what terminology we use to talk about radical expressions. So here is a radical expression. Looks pretty messy, but we're not going to evaluate it. We just want to break down some of these pieces.
Well, first we have the radical sign. So this lets us know that we're dealing with a radical. Then here we have the index. So the index here is 5. So it indicates the type of root. This is a 5th root. So our square roots have an index of 2. Our cube roots have an index of 3. Our n-th roots have an index of n.
Then we have what we call the radicand. And this is the expression that's underneath the radical sign. So 8x plus 5 is the radicand of this radical expression.
Now I'd like to talk about perfect squares and perfect cubes. And this is really going to help you simplify some radicals if you ever come across perfect squares and perfect cubes. So what I mean by perfect squares is let's write out a couple of integers-- 1, 2, 3, 4, 5, 6. And I can write on more and more. And I'm going to square these integers.
So 1 squared equals 1. 2 squared equals 4. 3 squared equals 9. 4 squared equals 16. 5 squared equals 25. And 6 squared equals 36.
Well, these numbers here in red are called perfect squares. And if we take the square root of these perfect squares, we're going to get an integer. Square root of 1 is 1. Square root of 4 is 2. Square root of 9 is 3. So on and so forth. So if you ever encounter a perfect square underneath a radical sign, it evaluates to an integer. Makes it easy to do in your head.
A similar idea exists with perfect cubes. So let's write out a couple of integers and cube them. So we have 1 cubed, 2 cubed, 3 cubed, so on and so forth. I can keep on going if I wanted to. Now, when we cube them, we get 1, 8, 27, 64, 125, 216, so on and so forth.
Now, these perfect cubes, when you take the cube root of them, you're going to get these integers. So the cube root of 1 is 1. The cube root of 8 is 2. The cube root of 27 is 3. So on and so forth. So if you ever encounter a perfect cube alone underneath a radical, that evaluates when you take the cube root into an integer.
Now, what if we don't have a perfect square or a perfect cube underneath our radical? So here we have the cube root of 30. 30 is not a perfect cube, so this is not going to be an integer. So this expression is the exact value for the cube root of 30. If you were to type this into your calculator, you would get a whole bunch of decimal digits. So this is the exact expression.
Now, if we wanted to approximate it using our calculator, there are a couple of things we could do. You would get 3.107, which is the approximation. To use your calculator, you can type in the radicand 30, and then your calculator might have a cubed root button. So you just type in 30 and press this button. And it will return the decimal approximation.
Other ways you can use your calculator is you might have to type in that radicand 30. And this then is the general radical button. And the next thing you have to type in is the index. So this is a cubed root, so you type in the 3. And this is another way to say the cube root of 30.
Or if your calculator does not have a radical button at all, you can use a fractional exponent. You can take your radicand 30 and raise it to a power. And because we're dealing with a cubed root, your power would be 1 over 3.
So what did we talk about with radicals? Well, we talked about the general radical, the n-th root of a. And the index-- n in this case-- indicates the type of radical. The radicand in this example is a. This is the expression underneath the radical sign. And then we talked about how the inverse operation of radicals is the exponent.
We talked about perfect squares. These are integers squared. So when we take the square root of perfect squares, we get integers. And we talked about perfect cubes. These are integers cubed. So if we take the cubed root of perfect cubes, we get an integer. Well, thanks for watching this video on an introduction to radicals. Hope to see you next time.
