 Hi, this is a video about exponents and their properties as An introduction we use exponents to represent repeated multiplication of the same factor the basic operation As seen as three to the force power means we take three and multiply it with itself four times three is called the base and four It's called the exponent if no exponent is present and it's understood that the power of the exponent is one anything Any base raised to the zero power is equal to one And most importantly When you see parentheses around the base it does make a difference for instance I have negative three and parentheses all squared This means that I'm taking negative three and multiplying with itself twice negative three times taking negative Three is positive nine next. I have negative sign and then three is squared So your negative sign stays as it is and you're squaring the three only that means three times three Three times three is nine and it's negative an example one will compute each of the following negative seven all in parentheses squared What am I squaring I'm squaring the negative and the seven So negative seven times negative seven It's forty nine and part B. I have eight to the zero power Any base to the zero power is always one Part C. I have negative four squared I'm only squaring in the four. So you have your negative sign four times four Which is negative sixteen Next working with negative exponents if you have a base a represents any numeric base or any base could be a variable to a base raised to a negative power if it's in the Denominator of a fraction you should literally just move it to the top of the fraction and the power becomes positive If you have a to a negative power on the top of a fraction You just move that a to that negative power to the bottom Make the power positive Just because an exponents negative does not make the base negative negative exponent does not affect the sign of its base It affects what side of the fraction the base goes on So I have seven X to the negative second I'm gonna write this over one just so you can kind of see that it's a fraction or it can be written as a fraction All I have to do is to write this in terms of having no negative exponents Take the X to the negative second and move it to the bottom of the fraction. I Get seven over X squared Next I have one over X to the negative third You take that X to the negative third move it to the top of the fraction. It becomes X to the positive third Last but not least you have negative two all to the negative fourth power I'll write it over one just so we can see how it looks as a fraction I literally take the negative two to the negative fourth power and move it to the bottom Of the fraction The power becomes positive So your job now is to evaluate negative two to the fourth power It's positive sixteen Next we'll move to some of the properties of exponents starting off with the product rule Product rules used to simplify when multiplying two bases together that are the same So for instance, I'll start off by doing example three part a I'm just going to stick with the basic definition of exponents. I'm not going to use the product rule to simplify just yet So part a can actually be written as three squared Is three times three and three to the fifth is three times three times three times three Times three in total. This is three to what power Well, there's seven threes multiplied together. So it's three to the seventh There's an easier way to do this when you're multiplying Two bases together that are the same. You can just add the exponents two plus five gives you seven That's what the product rules telling me if you're multiplying and the bases are the same add the exponents So another way to do part a that would take a lot less work would be three squared times three to the fifth is three to the two plus five Or three to the seventh same answer less work So in part B, let's use that shortcut to save us a little bit of time x to the fourth times x to the sixth Yes, you do have bases are the same. They are being multiplied together. This is x to the four plus two Or x to the sixth Part C has a little bit more action going on with it. You have a two x squared y to the first power Negative three x to the fifth y squared. This is all being multiplied together So I'm going to group this and write this a little bit differently because you can write multiplication in the order You want without changing the value of the expression I'm going to stick with multiplying my two Times the negative three so multiply those coefficients together I'm going to put my x's together x squared times x to the fifth and put my y's together Y to the first times y squared Two times negative three is negative six x squared x to the fifth actually gives me x to the Two plus five and then you have y to the one plus two so negative six x to the seventh Y cubed that is the final answer now. We'll talk about the quotient rule The quotient rules used to simplify when you're dividing two bases together that are the same So we'll illustrate the quotient rule in part a X to the sixth that means x times x times x times x times x Times x x is multiplied with itself six times all over x squared Notice you can cancel out two copies of x on top With two copies of x on the bottom of the fraction leaving you with Just x to the fourth You could have easily obtained that power of four by subtracting six minus two That's what the quotient rule is telling you to do So here's how the quotient rule actually works if you have x to the sixth over x squared this is x to the Six minus two or x to the fourth Notice you started your subtraction from the six the bigger number What would have happened if we would have started the subtraction from the bottom from the two Well, you would have gotten x to the two minus six Since you started your subtraction from the bottom you would have to put your result on the bottom To get one over x to the negative four Which is really x to the fourth after you make it a positive power So the key thing with the quotient rule is Wherever you start your subtraction from which I recommend always do the bigger number. That's where you put your answer and part b Let's first do negative 15 divided by three. That's going to give me negative 15 divided by three is five and Then with the y's Ten's bigger than seven. So I have y to the ten minus seven Everything's in the top of a fraction So I have negative five y cubed and part C. I have 18 over three, which is six X to the ten minus five which belongs on top because the tens where I started from and on the bottom I have y to the seven minus four Seven's the bigger number. It's on the bottom. So that's why seven minus four has to go on the bottom This leaves me with six six to the fifth over y cubed and that's your quotient rule Thanks for watching