 What lies beyond the basic arithmetic operation of addition, subtraction, multiplication, and division? We'll make the following definition. If I take a factor a and multiply it by itself n times, I can rewrite this in shorthand notation as a to superscript n. We read this as a to power n and we incidentally define a to power 1 is equal to a. So for example, let's evaluate five to power two and two-thirds to power four. So we'll pull in our definition. Now five to power two says that we're going to take five and multiply it by itself twice. So five to power two is five times five, and that's the same as 25. Two-thirds to power four says that we're going to take four copies of two-thirds and multiply them together, which gives us 16 over 81. It's useful to be able to go backwards. We can express an exponential form two times two times two times three times three. So we'll pull in our definition and the thing to recognize here is that we can only reduce something to exponential form if we're multiplying the same things together. So these two's, there's three of them, so we can rewrite these as two to power three. We can't ignore the rest of the expression. This three times three, there's two threes, so I can write that as three to power two. Where do exponents fall in the order of operations? The actual answer is they don't. Remember, a to power n is shorthand for an operation. It's not actually an operation. And what this means is that wherever possible, we should rewrite this in longhand. So this seven minus three to power two, we'll pull in our definition for exponents. Three to power two is the same thing as three times three. And order of operation says do the multiplication first, that seven minus nine, and subtraction of integer seven minus nine gives us negative two. This idea that exponents are shorthand is very important for evaluating expressions with signed numbers. For example, parenthesis negative three to power four versus node parenthesis negative three to power four. My shorthand says that parenthesis negative three to power four is four copies of negative three multiplied together. That's 81. Meanwhile, this negative three to power four, the four is attached to the three directly. And so that gives us four copies of three with the negative out front, and that gives us negative 81.