 In this video, I want to begin chapter four. In our series in chapter three, we talked a lot about quadratic functions and their graphs and equations and inequalities. Now we're at a point where we want to move on to higher degree polynomials, cubic polynomials, quartic polynomials, quintic polynomials. These things are just higher degrees, right? What if you want a degree three, a degree four, a degree five polynomial? Now it's gonna be requisite for us that as we start working with higher and higher and higher degree polynomials, as so-called power functions, we need to be affluent with exponential expressions and particularly the laws of exponents. The primary three are actually these three right here. So when you're working with an exponent, if you have any positive number A and you raise it to the exponent M, what happens when you multiply it together two exponential expressions? So I should make mention of that. When you see an expression like A to the N, collectively we refer this to as an exponential expression. The number on the bottom is called the base of the exponential and then this right here is what we often call the exponent or sometimes it's called the power. And then when you put these things together, we call that an exponential expression. And so the laws of exponents are what we can do to algebraically manipulate, simplify and substitute exponential expressions. If you have A to the M and you multiply that by A to the N, so the base is the same but we have exponents here. Turns out that you can add the exponents together. So when you multiply together exponential expressions, you add the exponents. And the idea is kind of like the following. If I have an A squared and I times that by an A cubed, well what that means is you have an A times A and then you have an A times A times A. Dropping the parentheses I have one, two, three, four, five, A's where five is two plus three. That's the idea behind that. We add together the exponents. Now on the other hand, when you multiply exponentials, you add powers. What happens when you divide exponentials? Well, if you divide exponentials, then we're gonna subtract the exponents. And the idea here is the following. If I have A to the fourth over A squared, you're gonna have A times A times A times A over A times A. If you cancel the common A's, you're left with just two A's and that's four minus two. So you subtract the exponents when you divide. And then the third one is if you have an exponential then you raise that to a power, then you can actually multiply together the powers. So for example, if you had like A squared and you're gonna cube that, what that means is A squared, A squared, A squared. So the first one gives you two, the second one gives you two, the last one gives you two. You're gonna end up with A to the two plus two plus two, which is A to the sixth, which of course is A to the two times three power. And so if you know those three laws of exponents, you're probably gonna be good as gold as we start talking about power functions and then later on exponential functions as well. But there's some other properties you should know when it comes to the laws of exponents. If you have two positive numbers A times B and you raise it to the power N, you can distribute that power and actually get A to the N times B to the N. And the idea of course is you have A times A all the way up to A and you have B times B all the way up times the B. The thing is since multiplication is commutative, you could reorganize all these things, right? And I guess I should have written it as like, you have A times B, you have A times B, and then you get A times B. So the idea is you have all these A's right here, you can gather them together in front and then you put all of the B's in the back. Because our operation is commutative, we can reorder everything, it's associative, we can redo the parentheses, we can redo this. And so you can distribute exponents across multiplication. It's also true for division A over B raised to the Nth power is the same thing as A to the N over B to the N, like so. Now I should mention that if there's laws of exponents, that means there's also crimes of exponents. If you break such a crime, you have to go to exponential prison. It is not true on the other hand that if you take A plus B, I'm writing this in red so it emphasizes blood. So think of death when you do this. If you take A plus B, many students try to make the mistake that this is A to the N plus B to the N. They try to distribute exponents across a sum or a difference and that is not true. This is the, these are crimes of exponents and this right here is public enemy number one. This is probably one of the most common mistakes that any student will do in an algebra class is that they'll be tempted to distribute exponents across a sum of some kind. And that's completely false. Like for example, if we took two plus three and we squared that, the proper calculation would be two plus three, which is five, you square that and you get 25. But it might be tempting to distribute this, right? Two squared plus three squared, two squared is four, three squared is nine, four plus nine is 13. And last I checked, let me see the comprehensive almanac of numbers. Yep, 13 and 25 are different numbers. So distribution of exponents is not a valid thing. So you wanna watch out for that. That'll lead you to problems. So don't distribute exponents. It's a bad thing, naughty, naughty, naughty, doesn't work. Some of the other exponential laws that I should mention here, if you take one to any power, you're gonna get back one. And that's because one times anything is just that number. So if I take one times one times one times one, it's just gonna be one. Also, if you take any number to zero power, that's gonna be one itself. And this is the idea that if you take a number to the zero of power, that's gonna be the multiplicative identity. And that's mostly a consequence of this principle right here. If you take a to the negative n, that's gonna equal one over n right here. And so it's the idea behind that's the following. Taking negative exponents actually gives you reciprocals. If I take a to the n, and I times that by a to the negative n, that ought to equal a to the n minus n, which is equal to a to the zero. But on the other hand, if I take a to the n, and I divide that by a to the n, which is what this thing's doing right here, by law number b, this should also be a to the zero, because I'm subtracting the exponents, but it also should equal one, because if you take a number to divide by itself, you get one. And that's kind of explaining where these things come from. Negative exponents actually means division, it means reciprocals. In that vein, can we make any sense out of fractional exponents? Well, if you look at law number c right here, if I were to take something like a to the one over n, and I raise it to the nth power, this by law number c should be a to the n over n power, which would be a to the first, which is just a. And so what we see here is that a to the one over n power, whoops, should be the nth root of a, the number which would raise to the nth power gives you a. And then when you put these things together, a to the mn, this is gonna be a to the one over n, raised to the nth power, and if a to the mth power, and so a to the one over n of course is the nth root, so you get something like this. And so this gives us a way of trying to compute exponential expressions. And so I wanna just do a handful of these examples to finish off this video right here. So how does one do something like four to the three halves power? What does a rational exponent mean? Well, the numerator gives you a power, the denominator gives you a radical. So four to the three halves means you take the square root of four, and you're gonna cube it. The square root of four of course is two, and two cubed is equal to eight. Notice how we can do this exponent, this rational exponent without the need of a calculator. What about negative eight to the four thirds power? Well, same thing, the one third power means you're gonna take the cube root of negative eight, and you're gonna raise that to the fourth power. Now be aware that taking the cube root of a negative number is not forbidden. That's a problem for square roots and all even roots, but odd roots, there's no problem. Cube root of negative eight is actually negative two. And to verify that, be aware that if I were to take negative two and I cubed it, you would get negative two times negative two, which is four, times negative two, which is negative eight. So no problem with that whatsoever. You can take the cube root of a negative number. Well, then if you take negative two to the fourth, that's gonna end up with a positive 16. A negative two squared is four, and then four squared is 16. Those are some exponential calculations with specific numbers. We can also simplify exponential expressions, maybe involving variables like X and Y right here. So when you have a product inside of an exponent, you wanna distribute that exponent, like on the first part. So you get X to the two thirds times Y. We're gonna multiply that by then, you're gonna get X to the negative two to the one half power, and then you're gonna get Y to the one half power. Now, when you have a composite of exponents, you multiply them together by law number C. So you're gonna get X to the two thirds Y times that by X to the negative one, Y to the negative one, one half. And so then you add together exponents, which then gives us, we're gonna get X to the two, I'm just gonna lower, ignore this equal sign, we'll come to that one a little bit later. You're gonna get X to the two thirds minus one, and you're gonna get Y to the first plus one half. We have to add some fractions here, but two thirds takeaway one is gonna be negative one third. And then Y, if the exponent there one plus a half, that'll be three halves, like so. And so this right here gives us the simplified form. Now, some people insist that we should write things without negative exponents. So if you do see that instructions, if you ever see a negative exponent, this means you just put in the denominator. And so you're gonna get Y to the three halves over X to the one half right there, or one third, excuse me. We can write this without any negative exponents whatsoever. And so then the last one D, let's work that one out there. Let's distribute the exponent of one half. So you're gonna end up with the square root of nine, because one half just means the square root. You're gonna get, you're gonna cut the exponent of X by one half. So you get two halves. And then for the Ys, you're gonna multiply the exponents together. And so you get one sixth. This will sit above X to the one sixth, and then likewise, Y to the one half. So what's going on here, right? Because you have a fraction inside of the exponent, I'm gonna be taking nine X squared, Y to the one third raised to the one half power. And this sits above X to the one third, Y raised to the one third power. So like I mentioned the law earlier, if you have a fraction inside of a exponent, we can take the X one of the numerator and denominator. Then as we have a product, we can distribute this into each of the pieces. So we get nine to the one half, we're gonna get X squared to the one half, and we're gonna get Y to the one third also raised to the one half power. And then the denominator, you're gonna get X to the one third raised to the one half, and then you get Y raised to the one half. And so when you have exponents nest composed with an exponents there, you multiply them together and that's what we did here in this white step right there. Simplify anything that we can. The square root of nine is three. X to the two halves power will just be X to the first. We'll get Y to the one sixth. And then at the bottom, we get X to the one sixth and Y to the one half. As we have X's and Y's on top and bottom, we can subtract their exponents. So we're gonna look like three times X to the one minus one sixth power. And then Y to the one sixth minus one half power. You might need a common denominator to add those fractions together. X to the six over six minus one sixth. And then Y to the one sixth minus three sixth. Simplifying that, we're gonna get three times X to the five sixth power. And then the last one, we're gonna get Y to the negative two sixth. And while that is a perfectly good way of writing the answer, I'm gonna write each of the exponents as positive simplified fractions. So we're gonna get three X to the five sixth. You can't reduce the fraction five sixth, but two sixth becomes a third. And since it's a negative X, what we're gonna put in the bottom, you get Y to the one third power. And so this would be the simplified expression for that exponential expression we started off with.