Fractional Exponents

Oh, my goodness. What in the world does something like this mean? Two to the power of three quarters? Take a breath. Look, I've got a plan. I wouldn't bring you to a place like this without a plan. Let's look at something more familiar. Something simple, like three to the power of two or three squared? That is three times three, which equals nine. What else are we going to need to understand? We're going to need to know about square roots. The square root of a number is the number that when multiplied by itself gives us the first number. If three squared is nine, then the square root of nine is three. We can even write it like this, using what is called radical notation. This symbol is called the radical. It goes over the number that we're taking the square root of. The nine, which we call the radicand, is under the radical sign. The radical sign is also used for roots other than square roots. We'll talk about other kinds of roots in a minute. But you should know that there's a spot that shows what kind of root we're taking from the Vatican. In this case, since it's a square root, the number that goes here is a two, and this number is called the index. But since there's no number here, we know it's a square root. But just because we don't see the two there, it's still there, hiding, where it's always been like a tiny ghost. Square roots are so common though that we don't even bother writing the index. We ain't afraid of no ghosts. Just remember, if there's no index on the radical, that means it's a square root. Okay, three squared is equal to nine. The square root of nine is equal to three. The square root undoes the squaring, and the squaring undoes the rooting. The square root is like hitting the undo button on a square. But remember, the ghost of the index of two is always there, even if we don't see it. This will become important later. Now, let's look at this. Any number raised to the power of one is just that number. So what? Well, a half plus a half is one, and we're still just raising x to the power of one. That's okay. We know that when we multiply fractions with the same base, we can add the exponents. So let's work that backwards here. If we're adding the exponents, that must mean that we can multiply them. Do you see where we're going with this? A square is a number times itself. So x to the power of one half times x to the power of one half is the same as x to the power of one half squared. So we take the square root of the x to the power of one half squared. And since this is an equation, see the equal sign, anything we do to one side, we have to do to the other side. We need to get the square root of x on the left hand side of the equation. And there you are. The square root of a number is equal to that number raised to the power of one half. And turning this around, we see that a number to the power of one half is the same as the square root of that number. So we've solved the mystery of at least one fractional exponent. But remember, that ghost of the index of 21. It's still there. It turns out we can generalize this. Let's turn that two into and n. And we can say that the root of a number is equal to that number raised to the power of one over n. Isn't that amazing? This works for any root. And this gets us several things. First, if we want to use a calculator to solve a root, let's say we need to know the 15th root of 97. Most calculators don't have a 15th root button. But using our formula, we know that the number is 97 to the power of one 15th. We can enter the base, then press the cart key to start the exponent. Then, and this is important. Hit the open parentheses key. To enter the fraction, we press the one key, then the division key, then enter 15, then the closed parentheses key, and then hit the inter key. Inside the calculator, it calculates the fraction and then uses it to raise the base to that power and gives us the answer. On a computer and a spreadsheet, for example, we do a similar operation. We enter the equals key to start the calculation, then enter the base, then hit the carrot key, open the parentheses, type in 1/15, and close the parentheses. Hit the ter key and the spreadsheet calculates the answer. We can also understand how to deal with general fractional exponents, like two to the power of three quarters. We can use the same trick, enter the base number, then hit the carrot key. Open the parentheses, enter in 3/4, Then close the parentheses, then hit the inter button. To understand a bit about what's going on under the hood of this calculation, we can say two to the power of three quarters is the same as two to the power of one quarter plus one quarter plus one quarter. So multiply two numbers with the same base, we add the exponents. We can say that two to the power of three quarters is the same as two to the power of one quarter raised to the power of three, and we know that two to the power of one quarter is the same as the fourth root of two. Two to the power of three quarters is the same as the fourth root of two raised to the power of three. We could also evaluate this expression in the calculator. Finally, being able to convert radical expressions to exponential form can also help us to multiply radical expressions, such as the square root of a times the fourth root of A. We can convert these to exponential form A to the power of one half times a to the power of one quarter. We can add the exponents to get A to the power of three quarters. So this simple equation is a very useful tool and very much worth knowing.