Today, we're going to talk about radicals.
Radicals are an operation, like taking the square root or the cubed root of a number. So we're going to go over how a radical and an exponent are inverses of each other. And then we're going to go over the language used for radicals. And then we'll do a few simple examples.
So let's look at why exponents and radicals are inverse operations of each other. Let's say I have the square root of 3 to the second power. So again, because a radical and an exponent are inverse operations of each other, they're going to cancel each other out. And this is just going to give me 3.
This radical and exponent are inverse. They cancel each other out. And so we just end up with the number underneath. Let's look a little bit more closely at what these two operations mean.
So if I have 5 to the second power, I know that the 2 exponent tells me that I'm multiplying by 5 two times. The exponent tells you how many times you're multiplying in a chain of multiplication. So 5 times 5 is going to give me 25.
Now, if I look at my radical, let's say I had the square root of 25. So a radical when you're evaluating asks you to find the number that when you multiply two times is going to give you 25. So I'm trying to figure out what number, if I multiply it by itself two times, will give me 25?
And that's just 5. So again, we see that these two operations are inverse of each other. I started with 5 and I squared it, which gave me 25. But when I take the square root of 25, I'm right back at 5 again.
Now, this will work for other exponents and radicals, too. So let's say I have 5 to the third power. Again, the exponent tells me that I'm multiplying by 5 three times. So 5 times 5 times 5 will give me 125.
Now, there is a radical called the cubed root, which if I took the cubed root of 125 I'm asking myself, what number when I multiply by itself three times will give me 125?
And again, this number is 5. So we can see they're inverse of each other.
If I had 5 to the seventh power, that's going to give me 78,125. So this is 5 multiplied by itself seven times. And there's a radical called the seventh root.
If I took the seventh root of 78,125. Again, is just going to get me back at 5.
So square roots and cube roots are most common type of radical. You notice that a cube root has a 3 here as a superscript with the radical, but the square doesn't have anything. So a square root, you don't write the 2 here as a subscript with the radical. If there's no number, you just assume that it's a 2. So it's a square root.
And we have a general form for writing numbers as a radical and some language to go along with it. So the number that's the superscript with the radical, we call that the index. And the number that's underneath the radical, we call that the radicand.
So if we look back at our examples, the square root has an index of 2, the cube root has an index of 3, and the seventh root has an index of 7.
So here's a few other things that you'll want to know about radicals. The first is the idea of a radical expression.
So for example, if I had something like the fifth root of 3x plus 10, we call this a radical expression because of the radical sign.
Again here, the 5 is the index of the radical and 3x plus 10 is our radicand. So you can have a radicand that is not just a number. It can be some combination of numbers and variables. So this is a radical expression.
So let's look at the idea behind perfect squares and perfect cubes. Perfect squares are numbers, such as 4, 9, and 16. And perfect cubes include numbers such as 8, 27, and 64. So let's first look at what makes a number a perfect square.
4 is a perfect square because there is an integer that I can square and get the exact value of 4. So 2 times 2 is 4. 2 squared gives me 4. So when I take the square root of 4, I know that that's just going to give me 2.
9 is a perfect square because there's an integer that I can square and get the value of 9. That's 3. 3 times 3 equals 9, which means the square root of 9 is 3.
Similarly, the square root of 16 is just going to give me 4. And that's because there's a number 4 that I can square and get the value of 16. So since 4 squared is 16, the square root of 16 equals 4. Let's look at some perfect cubes.
The cubed root of 8 is going to give me 2. And that's because there's an integer 2 which I can multiply three times and get the exact value of 8. So 2 times 2 times 2 gives me 8. And the cubed root of 8 is equal to 2.
So there's an integer that I can multiply 3 times to get the exact value of 27, which makes 27 a perfect cube. That integer is 3. And so the cube root of 27 is 3.
And finally, the cubed root of 64 is going to give me 4. Again, because there's an integer 4 which I can multiply three times to get 64. And so since 4 to the third power is 64, the cubed root of 64 is going to give me 3. I'm sorry, is going to give me 4.
So clearly, not all numbers are going to be perfect squares or perfect cubes. So for example, if I had the square root of 21, there is no integer that I can square and get the exact value of 21. So 21 is not a perfect square.
So this square root of 21 is going to evaluate to be a decimal. And it is approximately 4.583.
Another example would be the cubed root of 10. 10 is not a perfect cube because there's no integer I can multiply 3 times and get 10. So this is going to evaluate again to be a decimal. And it evaluates to be approximately 2.154.
So when you are evaluating radicals and the answer does not evaluate to be an integer, depending on the problem it may be better to leave your answer in terms of the radical, like the cubed root of 10 or the square root of 21. But it may be better again, depending on what the problem is asking for you to approximate that radical using a decimal.
So let's go over our key points from today. As usual, make sure you get them in your notes so you can refer to them later.
Radicals and exponents are inverse operations of each other. The index of a radical is the number in superscript above the radical sign and the radicand is the number underneath the radical sign. Radicals can be used for any n-th root, And the n-th root of a quantity is a number that is used as a factor n times in a chain of multiplication to arrive at the value underneath the radical. And finally, a radical expression is just an algebraic expression with the radical sign.
So I hope that these key points and examples helped you understand a little bit more about radicals. Keep using your notes and keep on practicing and soon you'll be a pro. Thanks for watching.
The expression underneath a radical sign.