 One important type of algebraic expression is known as a radical expression. We use the following terms. In the expression a to power n equals b, we say that b is the nth power of a, and that a is the nth root of b. We also have some special terms. If n, our exponent, is equal to 2, we can use the term square. If n equals 3, we can use the term cube. So we might describe the relationship 5 to power 3 equals 125. So we'll pull in our definition. So in the expression a to power n equals b, b is the nth power of a. Comparing our expression to our definition, we can then say that 125 is the third power of 5. Similarly, if a to power n equals b, we can say that a is the nth root of b. And again, comparing our expression to our definition, we can say that 5 is the third root of 125. And again, we have our special case. If n equals 3, if our exponent is equal to 3, we can use the term cube. So instead, we could say that 125 is the cube of 5, or that 5 is the cube root of 125. It's important to notice that a number may have more than one root. So since 5 to power 2 equals 25, and negative 5 to power 2 also equals 25, then from the first, we can say that 5 is a square root of 25. But the second tells us that negative 5 is also a square root of 25. Now this raises a new problem. We like it when questions have a unique answer. So questions like what is 3 times 5, or how much will this cost, or what day is the test, or will there be cake, we like there to be a single specific answer. We don't like it when there's multiple answers. To avoid multiple answers to the same question, we define some notation. First, the principal nth root of a, written this way, is the number whose nth power is equal to a, where the number, well, it satisfies certain properties that are a little complicated and depend on which root we're taking. However, the principal root we most commonly encounter is the principal square root. And so we say that the principal square root of a, written this way, is the non-negative number whose square is equal to a. So earlier, we said that the square root of 5 is 5 or negative 5, but when we write this thing, that is equal to 5 only. And a useful idea to keep in mind, how you speak influences how you think. So when you see this symbol, it's important to read it as the principal square root of a. The use of this symbol leads to the idea of a radical expression. A radical expression is an expression that contains a radical, a root of some sort. So principal square root of 5 plus principal cube root of 18 is a radical expression. 1 over 3 plus principal square root of 8 is also a radical expression. Now, radical expressions, as long as they're defined, are always equal to some real number, and so we might be able to simplify them a little bit. So let's try and simplify principal square root of 35 to the second power. So definitions are the whole of mathematics, all else is commentary. So according to our definition, the principal square root of a is the non-negative number whose square is a. And so by definition, the principal square root of 35 squared is the non-negative number whose square is 35 squared. So remember, paper is cheap, so let's go ahead and write some things down. We want something squared to equal 35 squared. And so the question you got to ask yourself is, what can I put in the parentheses that will make this a true statement? And here's a useful insight into the mind of a mathematician. We're not that subtle. How is something squared to be 35 squared? Well, why not just put a 35 in there? The only trick lies in remembering what will actually answer the question. We need the non-negative number whose square is 35 squared. Well, the thing we're squaring here is 35, and since 35 is non-negative, then the principal square root of 35 squared is 35. Well, let's do something a little bit more complicated. Let's try and simplify the principal square root of 16. So remember, the principal square root of 16 is the non-negative number whose square is equal to 16. So we want something squared to be 16. Since we aren't born with the knowledge of the square roots of the whole numbers, we should use guess and check to start with. It's useful to remember in any real-world problems, guess and check is always an option. And in fact, in most real-world problems, guess and check is the only option. So we find 1 squared is equal to 1. So that tells us the principal square root of 1, the number whose square is equal to 1 is 1. Now, some of the value of guess and check involves the computations we do along the way. In this case, we found the principal square root of 1. So if we ever need it, we already know what it is. Now, it's not worth memorizing. The square root of 1 is equal to 1. Because in practice, after we do this a couple of times, we just kind of remember it because we've done it before. Now, while we did find the principal square root of 1, that wasn't what we were looking for. So let's try another guess. 2 squared is equal to 4. And so that tells us the principal square root of 4 is equal to 2. Again, we want the principal square root of 16, so we're not there yet. Well, let's try 3 squared, which is equal to 9. So the principal square root of 9 is equal to 3. We can also look at 4 squared and 5 squared, which will give us the principal square root of 16 and the principal square root of 25. And the principal square root of 16 is the non-negative number whose square is 16, and we see here that 4 squared is equal to 16. So the principal square root of 16 is equal to 4. What about fractions? Well, here we can invoke a useful observation. The fraction a over b squared is a squared over b squared. And this means we can find the square root of a fraction by finding the square roots of the numerator and denominator separately. And this gives us a useful theorem. For a and b positive, the square root of the fraction a b-ths is the square root of a over the square root of b. So let's say we want to find the square root of 1625. So our theorem tells us that the square root of 16 over 25 is the same as the square root of 16 over the square root of 25. Now we need to find principal square root of 16 and principal square root of 25. So use guess and check and find a couple of squares. Or we could just go back to what we wrote earlier. And whether we recomputed these or just went back to our previous work, we see that the principal square root of 16 is 4 and the principal square root of 25 is 5. Equals means replaceable. So principal square root of 16 can be replaced with 4. Principal square root of 25 can be replaced with 5. And so square root of 16 over 25 is the same as the fraction for fifths.