 So how do we solve equations that involve a power or a root? A power equation is an equation of the form something to power n equals a, where our something is some expression involving a variable, and a is any real number. For example, x plus 7 to the 5 equals 8. Well, that's a something to the fifth equals number. That's a power equation. x cubed plus 7x plus 9 quantity squared equals 15 squared. Over on the left-hand side, we have something to a power. Over on the right-hand side, we have a real number. So this is a power equation. x cubed equals 5x plus 7. Well, over on the left-hand side, we have something to a power. But on the right-hand side, we don't have a real number. So this is not a power equation. And the important idea here is that we can solve power equations by taking the nth root. So let's take a closer look at that. Remember our definition of nth root. If b to power n equals a, then b is an nth root of a. So if we compare this to our power equation, x to power n equals a, then x is an nth root of a. And so we can write down one solution. x is the nth root of a. It's important to remember that when we write this symbol, we're referring to the principal nth root of a. And there may be others. So if x to power n equals a, then x equals the principal nth root of a is one solution. But there may be others. For example, suppose I want to solve the equation. They are a x plus 2 quantity squared equals 25. Since our equation is 3x plus 2 squared equals 25, if we ignore the things inside the parentheses, what we have is something squared equals 25. And so it follows that our something is a square root of 25. Well, we've already found these square roots of 25, 5, and negative 5. And so our something is either equal to 5 or it's equal to negative 5. At this point, we should apply the rule we learned in kindergarten, put things back where you found them. Originally, the parentheses held a 3x plus 2. We should put that back. And in fact, that 3x plus 2 should go in every set of parentheses. And now we have two equations that we can solve. So solving the first equation gives us a solution of x equals 1. And solving the second equation gives us a solution of negative 7 thirds. And so this original equation has two solutions. x equals 1, x equals negative 7 thirds. How about something like find a solution to quantity 3x plus 7 to the 5th equals 18. And we might proceed as follows. Since we now have an equation of the form something to the 5th equals 18, then something is a 5th root of 18. Now the problem only requires us to find a solution. So one solution is something is equal to our principal 5th root of 18. So that gives us our starting point. 3x plus 7 is equal to the principal 5th root of 18. And now we can solve this equation. And this gives us one solution. But it's important to remember that there are going to be other solutions. And a good math student, and a good human being, doesn't stop with the first solution that presents itself but looks for all solutions. Well, we're not yet in a position where we can find all solutions to this particular equation. But if you continue to take mathematics courses, you will find in a later course how to solve this type of equation. At this point, the only type of root for which we know all values for is the square root. Remember, if b is a square root of a, then minus b is also a square root of a. So let's focus on an important type of power equation, the square of a binomial. Ax plus b quantity squared equals c. So remember if b squared equals a, we say that b is the square root of a. So here we have something squared equals c, and so our something must be the square root of c. However, we can't write this as principal square root of c, since this is only the non-negative root. In particular, what that means is we'll get one solution, but there may be others. A good math student and a good human being doesn't stop with the first solution that pops into their head. They look for others, and in this case the thing to remember is that if b is a square root of a, then minus b is also a square root of a. So that means we have to write ax plus b equals plus or minus the principal square root of c. This is going to give us two equations. Ax plus b equals the principal square root of c, or ax plus b equals minus the principal square root of c. And so now we have two equations which we can solve separately. For example, you're walking along the street and out of the sky the equation 3x plus 5 quantity squared equals 36 falls and hits you on the head. And you say, whoa, this is square equal to number. I can solve this. So we know that 3x plus 5 is a square root of 36. We can write this as an equation. 3x plus 5 is plus or minus the principal square root of 36. The square roots of 36 are 6 and negative 6. So this gives us two equations which we can solve. And this gives us our two solutions. We should check our answers. Substituting x equals 1 third into our original equation, which is a true statement. So x equals 1 third is a solution. Substituting x equals minus 11 thirds into our original equation, which is also true. So x equals minus 11 thirds is also a solution. How about the equation 2 times x minus 4 squared plus 7 equals 25? Well, because this equation is in the video marked power equations, it must be a power equation. Well, not quite. Remember a power equation is an equation of the form power equals number. But the type of expression is determined by the last operation performed. And what that means is that while we do have a power over on the left-hand side, the expression is not a power, it's a sum. What can we do? Well, we can always try a little bit of algebra. Because we have a sum, add 7, we can get rid of the sum by subtracting 7. And now we still don't have a power equation because the expression on the left-hand side is not a power. The last thing we do on the left-hand side is multiply. The thing on the left is a product. Well, again, we can do some algebra. This is a product 2 times something. So we can get rid of the product by dividing by 2. And, well, we probably still don't have a power equal to number. So we look at the left-hand side and, oh, wait a minute, it is power equal to number. So this is now a power equation. So our equation is square equals 9. And so we can rewrite this as x minus 4 equals plus or minus the principal square root of 9. And since the square root of 9 is 3, that says that x minus 4 is equal to 3, or x minus 4 is equal to minus 3. And we can solve both equations.