 So what about the arithmetic of exponential expressions? To begin with, it helps to remember what our definition of exponents is going to be a to the power n is the product of n factors of a. And again, the useful thing to remember is that a to the power n is shorthand and it's not actually an operation. So if I want to find 2 to the power 5 times 2 to the power 3 and write the answer to exponential form, what I have to remember is 2 to the power 5 is 5 copies of 2 multiplied together. And likewise 2 to the power 3 is 3 copies of 2 multiplied together. And so when I multiply them all together, what I have is 5 plus 3, 8 copies of 2. And since I want to write the answer in exponential form, I can write this as 2 to the power 8. And it's a useful rule of exponents. A to the power m times a to the power n is a to the power m plus n. Notice that the base of the exponential expression a remains unchanged while the exponent itself is the sum of the individual exponents. So if I wanted to find 5 to the third times 5 to the tenth or x to the third times x to the tenth, so we'll pull in our theorem. So we'll keep the same base 5 but add the exponents 3 plus 10. And while it's useful to remember the rule, it's just as important to remember why the rule works. This 5 to the power 3 means I have 3 copies of 5 here, I have 10 copies of 5 here, and so when I run them all together I have 3 plus 10, I have 13 copies of 5. And it's important to remember that nothing important changes if instead of 5 we write x. x to the power 3 times x to the power 10 is the same as x to the power 13. So again while the theorem is useful, it's not actually necessary. And in fact we can just get rid of it as long as we remember what exponential notation really means. So here I want to rewrite in exponential form 2 to the fifth, 5 to the second, 2 to the fifth, 5 to the third. Because an exponential expression is simply a multiplication, we can rearrange it in any order that we want. So let's rearrange it so the 2s and the 5s are next to each other. Then we see here that we have 3 factors of 2, 5 more factors of 2, and so all together we have 8 factors of 2, which is 2 to the power 8. Likewise, this tells us we have 2 copies of 5 and 3 more copies of 5, which is 5 copies of 5, 5 to the power 5. And again nothing important changes if we use x's and y's instead of 2's and 3's. So here we have 3 and 5, 8 factors of x, and 2 and 3, 5 factors of y. What about quotients? What about 3 to the sixth over 3 to the second? So again it helps to bring in our definition of what an exponential expression is. The numerator, 3 to the power 6, that's 6 factors of 3. The denominator, 3 to the second, that's 2 factors of 3. We can cancel out some of the common factors of 3. We have 4 factors of 3 left over, which we can write an exponential form as 3 to the power 4, and this suggests a new rule for working with exponents when I have the quotient a to the power m over a to the power n, that's a to the power m minus n. So I can find the quotient 10 to the 8th over 10 to the 3rd, and I really only need my definition of exponents. I have 8 factors of 10 in the numerator, 3 factors of 10 in the denominator, and those 3 will cancel out with the 3 in the numerator, leaving us with 8 minus 3, 5 factors of 10. You should always be able to work from the definition, but it is time-consuming, and theorems are a good way to save time if you understand what they're saying. So let's put in that theorem, and so if I want to find x to the 8th divided by x to the 3rd, that's x to the power 8 minus 3 x to the 5th. How about a power of a power 5 to the 3rd to the 4th? So a good way to start is to pull in the definition, and that tells us 5 to the 3rd to the 4th is going to consist of 4 copies of 5 to the 3rd. And remember that the exponent tells us how many copies of the base. So here I have 3 copies of 5, 3 more copies of 5, 3 more copies of 5, and finally, 3 more copies of 5. So altogether I have 3, 6, 9, 12 copies of 5, and I can write that as 5 to power 12. Or you can use the theorem, but we didn't really need it. This does give us a new useful result. If I raise an exponential expression to a power, then because I have n copies of m factors of a, then I'm going to have a total of m times n factors of a. So if I want to find 10 to power 3 to power 5, that's 10 to power 3 times 5, 10 to the 15th. And again, we don't really need the theorem because this is 5 factors, each of which has 3 tens, so we have a total of 5 times 3, we have a total of 15 tens. And likewise, x to the 3rd to the 5th is going to be x to the 3 times 5, x to the power 15. Again, nothing really changes if we use variables instead of whole numbers. And nothing really changes if we use horrible numbers instead of nice numbers like 10 or square root of 11 over 6. So the hardest way possible to do this is to multiply these two numbers out and then do a lot of extra work. But we just want to write this in exponential form, and that means we just want to know how many copies of each factor we have. And so if we write this out 3,497 times 6,934 to the 3rd, well, that's three copies of this set. And multiplication is commutative and associative, so I can rearrange things. And exponential notation keeps track of how many of which factor. So here I have one, two, three factors of 34,97 and one, two, three factors of 69,34. And this suggests another rule. If I raise a product or a quotient to a power, then I get that many of each of the terms of the product or quotient.