 One of the important skills for success in pre-calculus is learning how to work with exponents. Now, a disclaimer, this video is a review of basic exponential expressions. It assumes that at some point you actually learned basic exponential expressions, but you need a little bit of a refresher. Maybe it's been several years since you've had to deal with exponents. And so, because of that, it consists of a set of rules per operating with basic exponential expressions followed by a couple of examples. This is by far the worst way to learn mathematics. You cannot learn mathematics well by memorizing a set of rules and following some examples. If you haven't already learned at some point how to operate with basic exponential expressions, you won't learn by watching this video. Instead, I suggest you watch some of the following videos, which I'll hyperlink to. That being said, let's take a look at a quick summary of our rules for exponents. In general, if I have a number a greater than zero, then my exponential expressions are going to be the following. If I have a to the power m times a to the power n, if I have a product of two expressions, then I can add the exponents. If I have the quotient a to m divided by a to the n, I can subtract the exponents. If I have a product of powers raised to a power a to the m, b to the n raised to power r, that's going to be a to m times r, b to n times r. That r outside exponent is multiplied by the two inside exponents. Roots, I could handle as fractional exponents. The nth root of a is the same as a to power 1 over n. Again, this requires a being greater than zero. And finally, we make two other identifications. First of all, a by itself, I can write it as a value with exponent 1 and likewise, a to the power of minus n that is the same as 1 over a to the n. And then finally, a to the power of zero is going to be equal to 1 as long as a is not equal to zero. Again, most of our rules of exponents, we assume our base is going to be positive. If our base is negative, we have to do some things with the signs and many of these things, many of these rules don't really apply. All right, so let's try an example. Let's say we want to rewrite an expression involving roots and negative exponents. And let's see if we can rewrite it without the roots and without the negative exponents. So roots are equivalent to powers. And in the order of operations, powers come first. So what I have here is I have a root and I'll convert those using the rule the nth root of a is the same as a to the power of 1 over n. So let's take a look at that. I have the cube root of x to the 5, y to the minus 7. That's x to the 5, y to the minus 7 to the power of 1 third. I have the square root x to the minus 3, y to the 3. So that's x to the minus 3, y to the 3 power of 1 half. So now I have an expression of that form, which eliminates the roots. Now I can also use the rule if I have a power of a power, I multiply the two exponents. So here I have x to the 5th, y to the minus 7 to the 1 third, x to the minus 3, y to the 3rd to the 1 half. And I'm going to multiply the 1 third by each of these exponents, the 1 half by each of these exponents. And so that'll give me x to the power of 5 thirds, y to the power of minus 7 thirds. This to the power of 1 half, I'm going to multiply it negative 3 halves and positive 3 halves. And now I have a product, so I can reorder the product any way I want to. And for convenience, I'll put all of the x exponents together and the y exponents together. And I have the product rule for exponents. If I'm multiplying two things with the same base, the product is going to be the sum of the two exponents. Now I have to subtract 5 thirds minus 3 halves. I have to add 7 thirds plus 3 halves and I'll do a little bit of fraction arithmetic. 5 thirds minus 3 halves is 1, 6. 7 thirds plus 3 halves is minus 5, 6. And there's my exponential expression so far. And this negative exponent, I do not want to have negative exponents. So my last step is the application of the rule a to the power of minus n is the same as 1 over a to the n. So I can take this last expression here, y to the power of minus 5, 6. And that's 1 over y to the power of 5, 6. And I'll do a little bit of cleanup on the rational expression there and multiply the two together. And here's my final answer. We can do some other things as well. So for example, here's a nice exponential expression. And maybe I want to rewrite this so I don't have parentheses and I have whole number exponents only. Now a useful thing to remember here is that in any fraction, the fraction bar, the line, is actually a grouping symbol. And essentially we throw parentheses around what's up top. We throw parentheses around what's bottom. And what that means is that we should consider each of these separately. Now the numerator, there isn't really much we could do with it. The denominator on the other hand, x, y squared, squared. And that is something we should take care of because that's a power and we can use our power rule for exponents. The denominator is going to be the square of an expression. First of all, I'll use x as x to the first. And now I can multiply that exponent in this x power 1 times 2. That's x squared. This y squared squared. That's 2 times 2. That's y to the fourth. And I have my first step in the expansion. Now I have a quotient, something divided by something. And I can, first of all, I can break it apart using my product rule for fractions in reverse. If I have a product of two fractions, I can combine them. And I can also go in the other direction. So that splits. And now I can apply the quotient rule for exponents. If I divide two exponential expressions, the result is the difference of the exponents. This is x cubed over x squared. This is x to the power 3 minus 2. And y to the power 2 minus 4. And I'll clean that up. 3 minus 2 is 1. 2 minus 4 is negative 2. And finally, I want to use the rule a to the power 1. It's the same as a. A to the power minus n is the same as 1 over a to the n. So this expression here, this is just x. This is 1 over y squared. And again, I can do a little bit of cleanup of the fractions. This is x over y squared. One more example, maybe I want to rewrite without using rational exponents. So let's go ahead and take a look at that. Again, I have a thing raised to a power. And so I can use the power rule that says that if I do that, my new exponents are the product of the two exponents. So I'll first multiply this exponent 4 by 1 third to get 4 thirds. 4 by negative 3 fifths gets me negative 12 fifths. Now, how do I handle that? Well, first of all, I can eliminate the negative exponents using the rule a to the power minus n is 1 over a to the n. So that's gone. And those fractional exponents, I'm going to combine two different rules. One is a basic rule of fractions. A over b is the same as a times 1 over b. And my rules of exponents say that a to the power n n is the same as a to m raised to the power n. And what that allows me to do is that allows me to split the fractional exponents into a whole number portion and a 1 over n portion. So this x to the power 4 thirds, that's x to the fourth to power 1 third. This is y to the 12th to power 1 fifth. And so that'll split like that. And finally, I can use the rule that says radical expressions, roots are going to be the same as 1 over nth powers. So this is a 1 over 3. This is a cube root 1 over 5. This is a fifth root. And my final expression looks like that.