hi this video is going to cover exponent rules so the learning goals for this video are to develop and understand exponent rules and also to learn how to apply the rules uh to simplify expressions with exponents um so we're just going to go through some terminology here here we have an Expression 2 to the 5 that five here that is what we call an exponent that big two is what we call a base and as a whole 2 to the 5 is what we call a power so we're going to be using these words uh so hopefully it's clear uh what each of those words mean all right we're going to simplify this expression here we have 5 to the 5 uh * 5 to 3 well 5 to 5 means this 5 * 5 * 5 * 5 * 5 there's five of them and 5 to the 3 means uh this 5 5 * 5 or three times um now those two are multiplying so we have this long string of fives now if you count there's eight of them okay so if we are multiplying five here okay eight times well well that's just uh five to the 8 now if you notice here uh in the beginning we have a five and a three and at the end we have an eight well five + 3 gives me 8 and again there's eight of these numbers right where we're just basically putting um uh these five fivs and these three fives together okay to make eight of them so when we're multiplying powers that have the same base we can add the exponents as a shortcut uh now let's take a look at what happens if we divide these things so again 5 to the 5 means that if we're dividing with 5 to the 3 okay that's what we mean now bed Mass tells us that we can divide and multiply in any order we want uh so I'm just going to divide five with five and five divide five five gives me one I can do the same thing again that'll get me one and then this five and this five divided together will get me one so let's just look at what we have left here we've got 1 * 1 * 1 which is 1 Time 5 which is five Time 5 again which is 25 so Al together right now we have 25 and 25 is the same thing as well 5 * 5 which is uh better written as 5^ squared so if we take a look at the beginning here we had a five and a three as exponents and we're ending up with two um well five subtract three gives me two so when we're dividing powers that have the same base all we need to do is subtract the exponents and um we're basically taking away three of the fives okay because they were three on the bottom here so we're just left with two so these are uh two very crucial rules about multiplying and dividing powers that have the same base and that's very critical um let's look at another rule we have um again a sar remember that's just um yeah just different color here that's just a * a right okay well this just a squar um now if we have 2 3 squared well that just means 2 to the 3 * itself so 2 the 3 * itself so it looks like this now just from before second ago if we have powers at the same base and they're multiplying well then the rule was we get to add these exponents so 3 + 3 gives me 6 also down here we have 5 4 raised again to uh a three so that just means 5 to 4 multiply by itself well three times and the rule again is that well when we have powers that are multiplying they have the same base we can add these exponents so 4 + 4 + 4 gives me a 12 uh let's look at the bottom two rows here uh a three and a two now all of a sudden turned into six and then a four and a three turned into a 12 now 3 * 2 gives me 6 4 * 3 gives me 12 so here we have another rule we're not adding or subtracting the exponents uh but when a power is raised to another exponent uh you can multiply the exponents so now we've got uh three rules one where we add subtract or multiply the exponents so the difficulty here is just making sure when to do what so let's just do some examples um so that we're a little bit more clear uh so here here um we're going to simplify each one x 4 * x 6 should get us well um they're the same base and they're multiplying so we get to add these exponents so 4 + 6 will get me 10 so that's x to the 10 uh here we have two uh powers that are dividing so we have t 8 / T now there is no exponent here but there is one um it's an invisible one it's an invisible one uh so when powers are dividing and they have the same base remember we can subtract these exponents so T 8 / t to the 1 essentially is T to the 7 we just subtract them uh here oh we've got two things happening there's multiplication or a power raised to an exponent so bed Mass tells us that we have to deal with the exponent before we multiply so let's do that we're going to leave this B to the 2 by itself first and then we're going to settle this so B to the 3 raised to the 5 well we can multiply these two exponents so 3 * 5 is 15 okay and uh now these two powers are multiplying so we can add these exponents so we get B to the 17 okay so we need to deal with the exponent first before we do any type of multiplying or in uh in this example here with d dividing so we need to deal with uh the power raised to an exponent so a 5 raised to a two again we get to multiply those exponents so that's a to the 10 divided by a and now again there is no exponent there but there is one there's an invisible one okay so it helps to write it uh to help visualize uh and a 10 / a 1 is well a to the 9 because here when we're dividing we get to subtract the exponents all right now let's be clear about uh the keys here we need to be clear about the exponent rules of course so let's just go over them again so when we're multiplying same bases we get to add exponents when there's division of powers with the same bases we subtract exponents and when there's a power raised to a power we multiply exponents okay and just like before if the expressions are fractions you want to simplify the numerator completely uh or the denominator uh fully and then uh divide the numerator and the denominator uh at the end okay so let's just um do some examples here you're going to give them a shot um so be clear about what rules need to be um used first okayy uh pause the video give these a shot and then when you're ready just press play good luck all right uh for question a here uh these two powers are multiplying so again we get to add these exponents right so that should be a to the 7 uh for B here uh the numerator uh the bases are sorry the powers are multiplying so we get to add that so that's X to 7 and then divide by x to the 1 we'll just put the one there to clarify and they're dividing so we get to subtract the exponents that should be x to the 6 try a couple harder ones again pause the video give it a shot good luck all right here we need to make sure that we deal with the exponent first before we multiply so this y to 4 should be on its own first we're going to deal with the power raised to an exponent so y 3^ 2ar we multiply those exponents so that gives us y 6 and then these two um powers are multiplying so we add the exponents so that's y the 10 all right this last one's a doozy we need to completely simplify the numerator and then also completely simplify the denominator and then divide both afterwards so the numerator here it's a power raised to an exponent so we get to multiply those um exponents so that's D to the 4 * 3 which is 12 the bottom here the powers are multiplying so we get to add the exponents so that's D to the 2 + 9 is 11 now uh these two powers are dividing so we subtract so 12 take away 11 is 1 so that's D to the 1 but it's uh more common that we don't write the one it's just implied so we just write d right well I hope you got the those correct um so again uh the most difficult thing about this topic is when to do what so when powers are multiplying say x to 4 * x 3 okay remember we get to add these exponents right so that should get us X to 7 okay so that's add when we're dividing so X to 4 ided by x 3 remember we get to subtract these so that should be x to the 1 which is just X so when they're dividing we subtract and then finally a power uh to a power so let's say x to 4 raised to a 3 so 4 * 3 should be 12 okay so a power raised to a power here we do some multiplying okay so hopefully this video has made exponent rules uh clear as well as how uh we use them to simplify expression so um again I hope that's clear and best of luck to you