in this video we're going to talk about basic properties of exponents and we're going to make it where the exponents are just positive whole numbers um exponents can also be negative numbers they can also be fractions they can be decimals um they can be basically just about anything so but we'll start with the basic case on this one so all an exponent really is it's just a shorthand way of writing something so suppose I have 3 * 3 * 3 * 3 * 3 um well notice I have five threes every time I'm multiplying something by itself a rather large number of times I don't want to write it out every single time so the way we write this compactly is 3 raised to the 5ifth power so basically it says you have five or excuse me three multiplied by itself five times and this gives us one of our first basic rules it says if you have a number a raised to a power M and we multiply this by a raised to the power of n it says what we need to do to simplify that is to add the exponents so for example if we wanted to simplify say 3 to 4th * 3 the 2 we'll simply combine that and have 3 to the 6 and if you think about it this makes sense because 3 to 4th is 3 * 3 * 3 * 3 you've got four of those 3 2ar is 3 * 3 well if you count them up how many threes do I have being multiplied by themselves well I have six and that's what this rule says our next basic rule says that if you have a number raised to an m the m power divided by that same number raised to the N power and I'm going to box this one in up here before I forget so this is an important property it says you can simplify this by taking a and you subtract the exponents you take the top exponent and subtract away the bottom one so again to maybe kind of convince you this by example if I had 3 the 3 or let's make it 3 4th over 3^ 2ar well that means I have 3 * 3 * 3 * 3 I've got 3 * 3 on the bottom if I cancel two of those out what am I left with I'm left with 3^ squar on top which is the same thing I would have gotten if I had taken the power 4 minus 2 so this does seem to work out let's look at a few more basic properties of exponents here and then we'll move on to some more complicated examples suppose I have well the next property and then we'll do an example it says if you have a raised to the power of M and then we raise that to the power of n this time you multiply those two Powers M and N together so suppose I have 2 to the 3 raised to the let's say oh second power well remember I've got something squared that means I have it multiplied by itself twice and 2 to the 3 power is just a shorthand way of writing 2 * 2 * 2 I've got another one 2 * 2 * 2 well how many twos do I have all together I've got six of them that are being multiplied all together and that's the same thing had we just multiplied the powers at the very beginning we would have also gotten 2 to the 6th power okay so just another little basic property here and a variation on this is what if I have say a * B raised to the m power in this case if you have two things in the parentheses that are being multiplied you'll take the first one and raise it to the power and multiply that by the second one raised to that power this is another basic exponential rule that you'll want to know and to maybe convince you that this is in fact valid suppose I have two X raised to the 3 power well again that means I have 2X * 2x time 2x I've got three of those because again it's being raised to the third power I can reorder this since this is all multiplication I can re you know just again reorder it so I can make this 2 * 2 * 2 x * x * X and I'm left with 2 the 3 power I have 3 tws being multiplied x to the 3 power I have 3 X's being multiplied as well so just another little basic rule that you'll want to know and honestly we're not talking about it but these rules will hold it doesn't matter whether we're using positive exponents whether we're using negative exponents um um whatever the power is this is going to be true last little case um and it's just a variation on this one suppose I have a / B raised to the m power well it says in that case you're going to get a raised to the m power over B raised to the m power and hopefully I can convince you with an example just like the last one suppose I have this time 2 / X raised to the 3 power well that means I have 2 overx * 2x * 2x well I've got 2 * 2 * 2 on top that's 2 the 3 I have x * x * X on the bottom that leaves me with x the 3r obviously you could simplify Down 2 the 3 and get just plain old eight but we'll just leave it like it is for now again just to illustrate this this power or excuse me this property so let's do a couple more complicated examples kind of combining these rules and we'll do a bunch of them here so suppose I have 2 rais to the 4th raised to the 3 well this is the property where if it's in the parentheses I just multiply so I'll get 2 12th and if you wanted to you could multiply that out 2 * 2 * 2 12 times but I don't want to and I don't have a calculator in front of me so we'll just leave it just like that suppose I have oh sorry somebody's calling me here let's turn that off suppose I have 2X raised to the thir power well again each thing gets the power so that's 2 the 3 x the 3 and in this case I will simplify it Down 2 the 3 that's 2 * 2 * 2 that'll give me 8 and x to the 3 I can't really simplify that down so I'll just leave it just like it is all right let's keep going here suppose I have x to the 4th raised to the 3 power over X2 well you have to be careful you can't just cancel things out immediately because the thing in the denominator it's not the whole thing being raised to the third power only the X to the 4 but if I simplify that if I use the multiplication property 4 and 3 is 12 over X the second and now the idea is I can subtract exponents so top minus bottom that's 12 Min - 2 that'll leave me with X to the 10th power let's do some more of these here suppose I have y to 5th raised to the 3 power y the 3 power squared and suppose that's divided by y 4th raised to the 4th power well again if we simplify this I'm going to simplify the first part so if I multiply these two numbers the five and the three I'll get y to the 15th I'll get a y to the 6th power on the other part if I use my multiplication property in the denominator 4 * 4 is 16 and now there's a bunch of ways you can simplify this I'm going to do the stuff in the top I'll get y to the 21st power 16 is still hanging out on the bottom but now I can use my cancellation property as before so if I take 21 minus 16 I'm going to be left with Y to the 5th whoops got a little cut off there y the 5th is my solution let's do one more of these suppose I have X2 y 3 over let's say x to 4th all raised to the fifth power well the first thing I'm going to do on this one since it's all raised to the fifth power I'm going to simplify inside the parentheses so let's see actually yeah this is okay so I've got y to the 3 on top oops wrote y to the 5th so if I take the the top power actually this one's going to be maybe a little trickier so let's think about this as being X * X on the bottom I have four X's I can actually cancel out two of the x's on top so that I'm left with Y the 3 on the bottom I have X squ still in the bottom again I've got two of them raised to the fifth power and again now if I use my multiplication property I just take the three and the five and get y to the 15th I do the same thing on the bottom and I'll get x to the 10th power and that'll be my final answer in this case so here are some basic properties of exponentials again you can use fractional exponents you can have negative exponents um and I'm definitely going to do some more examples of those in some other videos so definitely take a look at my website and you can find some other more complicated examples if these are a little too easy for you