 So what if we have an equation that includes a root? In general, we can solve a root equation of the form nth root of expression equals a using the same relationship we used to solve a power equation. From the nth root of x equal to a, we have a to power n equals x. Except there's one important change. Remember that when we write this notation nth root of x, we're referring to the principal nth root. On the other hand, we can raise any number, positive or negative, to the nth power. And this means it may be possible to have solutions to the equation a to power n equals x that are not solutions to nth root of x equal to a. These non-solutions are called extraneous solutions and should never be included as solutions. And so there are three things that are very important to remember about solving root equations. Because we have to worry about extraneous solutions, always check to make sure your solutions satisfy the original equation. Second, because there are extraneous solutions, always make sure your solutions satisfy the original equation. But possibly the most important thing to remember about solving root equations, always check to make sure your solutions satisfy the original equation. So, for example, let's take the equation square root 3x plus 7 equal to 5. So we can begin with the fact that if the nth root of x is equal to a, then a to the power n equals x. So if square root of 3x plus 7 is equal to 5, then 3x plus 7 itself is equal to 5 squared. Well, that's a nice, simple, linear equation that I can solve. And again, x equals 6 as a solution. So remember the three important things to remember about root equations. So we need to check if x equals 6 actually solves the original equation. So we'll substitute 6 into our original equation and see if we get a true statement, which we do so we're confident in writing down x equals 6 as our solution. Well, let's take a look at another one. Now remember, definitions are the whole of mathematics, all else is commentary. And if you really take this to heart, you have an easy, fast solution to this problem. And that's based on the definition of the symbol square root, which is the non-negative number whose square is equal to a. And what that means is that square root of a always indicates a non-negative number, which means it's impossible for the square root of 3x plus 7 to be negative 5. And so we might say that since square root of x always gives a non-negative number, there is no solution to this problem. Well, maybe you don't like doing things the easy way and would much rather do things the hard way. So we can try to solve this and see what happens. So remember, if the nth root of x is equal to a, then a to the power n is equal to x. And so I could rewrite this equation as 3x plus 7 is equal to negative 5 squared. We'll solve the equation. And because this is a root equation, we need to check back in the original equation. I get square root 25 equals negative 5, but this is false. Because remember that the square root symbol always gives you a non-negative number. So this x equals 6, even though it solved this equation back here, it did not solve the original equation. And so it should not be included as a solution. Now let's spend a few seconds thinking about how we might summarize our results. The best way to summarize our results is also the easy solution, which is recognizing that square root of x must always be a non-negative number, so there is no solution. But if you don't like finding the easy solution, and in particular, if you've done all of this work, you might want to at least mention it. And so one way you might do that is the following. We rewrote our equation as 3x plus 7 equals minus 5 squared. And we found that x equals 6 solves this equation. However, x equals 6 does not solve the original equation, so this solution is extraneous, and the original equation has no solutions. But again, you're really better off if you immediately recognize that principal square root always gives you a non-negative number, so equations like this are inherently unsolvable. How about cube root 8 minus 2x minus 3 equals 2? The first thing to recognize is that this is not in the form n-th root of expression equals a, and so we should use some algebraic manipulation to get it into that form. So let's take a look at this. Over on the left-hand side, we have a subtraction, so we can get rid of that subtraction by adding. And now we have it in the form root equals number. So we could raise both sides to the third power because this is a cube root. We'll solve our equation, and we'll check our answer, which works out.