 One of the goals of algebra is to be able to solve every type of equation that exists. Well, we can't actually do that, but we can solve a variety of equations. And so after linear equations, the next important type of equation is a power or root equation. So we'll start with the following definition. If b to power n equals a, then b is an nth root of a, and conversely. Now, people being how they are, we tend to use special words for common cases. And the two common cases are the second power. If b to the second equals a, we should say that b is a tooth or a second root of a, but we actually say that b is a square root of a. And likewise, if b to the third equals a, we say that b is a cube root of a, even though we should say it's a third root. So let's try to find the square, the second root of 25. So we'll pull on our definition. Remember that the nth root of a is a number where b to the power n is equal to a. So since we're not born with the knowledge of all the square roots, let's make a few guesses and see where it takes us. Is it zero? Well, we'll check it out. Zero squared is equal to zero. So it's not zero. How about one? One squared is equal to one, not that. Two squared is equal to and we can continue to try other numbers. And we finally find our root 5 squared is equal to 25. Well, that's one square root. Could there be others? Remember that if 5 squared is equal to 25, then negative 5 squared is also going to be 25. So both 5 and negative 5 are square roots of 25. This does raise a complicated problem. We generally don't like getting two different answers to the same question. So to avoid that, we'll introduce the following notation. This notation is read as the principal nth root of a. The idea is that while we might have several nth roots of a number, we're going to select one of them to be the principal one. Which one? Well, in the case of the square root, the one we're going to select is the non-negative number whose square is a. So let's ask the same question with a slightly different format. First, find the square roots of 25 and also find, well, you might have read this as square root of 25 in previous classes, but how you speak influences how you think. So it's a good idea to read this as the principal square root of 25. So we've already found that 5 and negative 5 are square roots of 25. But when we write this symbol, the principal square root of 25, that has to be a non-negative number. And so this principal square root of 25 is going to be 5. Let's think of this a little bit further. Suppose b is a square root of a. Then we know from our definition b squared is equal to a. But now consider negative b. We have negative b squared equals, which is b squared, and since b squared is equal to a, then we have negative b squared is also equal to a. So that means negative b is also a square root of a. And this leads to the following result. If b is a square root of a, then negative b is also a square root of a. So now let's think about this in terms of equations. If x squared is equal to a, then x is a square root of a. Now we'd like to write this as a formula, but if we write x equals principal square root of a, we are choosing only the non-negative square root. It's important to remember that the symbol principal square root of a always indicates a non-negative number. But we'd like to have all solutions. So if we want to indicate all solutions, we must write x equals plus or minus the principal square root of a to include both solutions. And this is an important idea to understand. So if the statement is incorrect, fix it. Square root of 4 equals plus or minus 2. So the important thing to remember is that our square root symbol always indicates a non-negative number. So this symbol square root of 4 is the non-negative number whose square is 4. Since 2 squared is equal to 4 and 2 is non-negative, then square root of 4 equals 2.