do you know why i pulled you over today uh some law that's right allah this law to be exact laws of exponents i've never even heard of these how was i supposed to know well you must have never heard of math antics they're this really cool math video series they have all sorts of basic math videos and the host is really funny is this some sort of ad can i skip this no i think you'll really like it check it out hi this is rob welcome to math antics in this video we're going to learn about the laws of exponents if you look up the laws of exponents online or in a math book you'll probably see a long list of equations that look something like this wow that's kind of overwhelming when you see them all at once but don't worry we'll take them one step at a time and you'll see that they're not that complicated after all but before we get going if you're not confident with the basics of exponents i highly recommend watching our previous videos about them before moving on in this video okay let's start with just the first two laws on our list which should look familiar if you watched our video called exponents in algebra they're just the two rules we learned in that video you know rules laws same difference so you probably already know them they simply tell us that anything raised to the first power is itself and anything raised to the zeroth power is just one and because we already know about exponents that are higher integer values like x to the second x to the third that means we've pretty much got things covered right ah not so fast don't forget that integers can have negative values too for example what if we add the expression x to the power of negative one or negative two or negative three we know that exponents are a way of doing repeated multiplication but how in the world could you multiply something together a negative number of times well you can't fortunately the next law on our list tells us how to interpret a negative exponent that law says x to the negative nth power equals 1 divided by x to the nth power and if you think about it that kind of makes sense a negative number is the inverse of its positive counterpart and division is the inverse operation of multiplication right so a negative exponent is basically repeated division x to the negative 1 would be 1 divided by x x to the negative 2 would be 1 divided by x divided by x x to the negative 3 would be 1 divided by x divided by x divided by x and so on seeing it like this makes the pattern clear but mathematicians prefer to express negative exponents in fraction form where one is divided by the same number of x's multiplied together but since those multiplied x's are all on the bottom of the fraction you're actually dividing by all of them here's an example that will help you see that that's true 2 to the negative third power let's first try that as a repeated division problem like our pattern shows us we always start with a 1 so we would have 1 divided by 2 divided by 2 divided by 2. if we do those operations from left to right using a calculator we get 0.125 as the answer now let's write it in fraction form like our law of exponents tells us we can 2 to the negative 3rd power would be the same as one divided by two to the third power and that's the same as one over two times two times two which simplifies to one over eight and one over eight simplifies to 0.125 see whether you write it as a pattern of repeated division or in fraction form like our inverse law shows you get the same answer and now you know how to handle any expression with a negative exponent it's just 1 over the same expression with a positive exponent x to the negative one is one over x to the positive one or just one over x x to the negative two is one over x to the positive two x to the negative three is one over x to the positive three and so on all right three laws down five more to go and these next five show us how we can do various math operations involving exponents in fact the next law tells us how we can take a number raised to a power and then raise that to a power as you can see it shows an expression x to the power of m grouped inside parentheses and then that whole group is being raised to the nth power it's a nesting situation kind of like those russian nesting dolls so what if someone asks you to simplify the expression x squared cubed which means the entire x squared term is raised to the third power well our law tells us that we can simplify that by multiplying the exponents together see how it equals x to the power of m n which means m times n that means x squared raised to the third power would be the same as x to the power of two times three which is six wanna see why that's true well think about what it would mean to raise x squared to the third power it would mean multiplying three x squared terms together like this and each one of those x squared terms simplifies to x times x right so we end up with six x's being multiplied together which is just x to the sixth power see our law works great if you have a number raised to a power and that's all raised to another power you can just multiply the two exponents together to simplify it and it works for negative exponents too like what if we had x squared raised to the negative third power well our law tells us that that's the same as x to the power of two times negative three which is x to the negative sixth to see if that's true we'll need to use the law we just learned about negative exponents and rewrite this as one over x squared to the positive third power that simplifies to one over x squared times x squared times x squared which in turn simplifies to one over six x's being multiplied together and that all checks out because one over x to the sixth would be the same as x to the negative sixth power pretty cool huh okay we're halfway through our list of laws and we're gonna look at the next two as a set because they tell us how we can multiply and divide expressions that have the same base and that's important because we couldn't simplify them to have a single exponent if the bases were different the first law says that if we have the base x with exponent m being multiplied by the same base x with exponent n we can combine them simply by adding the exponents together and the second law says if we have the base x with exponent m being divided by the same base x with exponent n we can combine them simply by subtracting the exponents let's see some examples of each like this one two to the third times two to the fourth does that fit the pattern of our first law yep the base of both expressions is the same but they happen to have different exponents the law would still work if the exponents were the same but they don't have to be just the bases have to be the same our law tells us that this would equal 2 to the power of three plus four or two to the seventh power but does it well let's break it down and see two to the third is two times two times two and that's being multiplied by two to the fourth which is two times two times two times two that's a lot of twos being multiplied together seven two's to be exact aha so that is what you'd get by just adding the exponents together since three plus four equals seven and this law makes total sense if you think about what an exponent really means the exponent is telling you to do repeated multiplication of the base right so this first part is telling you to multiply three twos together and the second part is telling you to multiply four twos together so that's why you can add the exponents together if the base is the same or think about it like this if we had 10 x's all being multiplied together we could form different groups of them and combine them using exponents like we could combine the first four x's into x to the fourth and combine the remaining six x's into x to the sixth and of course those expressions would be multiplied together since all of the x's were being multiplied but you'd probably never want to do that would you i mean why not just combine all 10 x's into the expression x to the tenth ah but there you see that our law holds true x to the fourth times x to the sixth would equal x to the power of four plus six or x to the tenth okay now let's move on and see some examples of the second law in this set which tells us how to divide expressions with the same base suppose we have the expression 5 to the third power divided by 5 to the second power our law says that we can simplify this by subtracting the exponents specifically we take the exponent on the top and subtract the exponent on the bottom from it if we do that the simplified version would be 5 to the power of 3 minus 2 or 5 to the first power but is that right to see let's write the expression out in expanded form on the top of our division problem we have five to the third which is five times five times five and on the bottom we have five to the second which is five times five does this look like something you've seen while simplifying fractions yep since all the bases are the same they form pairs of common factors on the top and bottom that can be canceled out this 5 over 5 cancels and this 5 over 5 cancels if you don't know why that works be sure to watch our video about simplifying fractions and what do we end up with well all the factors on the bottom cancelled out which leaves 1 since there's always a factor of 1 and there's only one 5 left on the top so our expression simplified to 5 over one or just five following our law we got a simplified version of five to the first power which is also just five so it really did work but to make sure you've really got it let's try using this law again with the expression x to the fourth power over x to the sixth power this one's interesting our law says that we can simplify it by subtracting the bottom exponent from the top right but in this case that will give us a negative exponent because the bottom exponent is bigger than the top four minus six would be negative two so according to our law this expression should be equal to x to the negative two let's try writing it out in expanded form to see if that's true on the top x to the fourth would be the same as four x's multiplied together and on the bottom x to the sixth would be the same as six x's multiplied together once again we see that there are pairs of common factors that we can cancel four pairs to be precise and when we cancel them all we're left with a one on the top because there's always a factor of one and only two x's multiplied together on the bottom if we recombine these two x's we get one over x squared and if you remember the law we learned earlier about negative exponents you'll see that one over x squared is exactly the same as x to the power of negative two so these laws really do work okay it's finally time to look at the last two laws on our list and fortunately they're pretty easy ones so we're not going to spend too much time on them these laws look kind of similar to the last pair just like before the first one involves multiplication and the second one involves division but notice that in these laws the bases are different but the exponents are the same that's the exact opposite of the situation with the last pair of laws it turns out that these laws aren't about simplifying exponents they're about how you can distribute or undistribute a common exponent to different bases the first law shows this group x times y that's being raised to the power of m and it says that you can rewrite it as x to the m times y to the m in other words you can distribute the exponent to each factor in the group and the second law shows the group x divided by y that's being raised to the power of n and it says that you can rewrite it as x to the n divided by y to the n in other words you can distribute the exponent to each part of the fraction of course these laws would work in reverse too and you could undistribute the exponents if they're the same for example if you're given the expression x squared times y squared you could rewrite that as the quantity x times y squared and if you have the expression x squared divided by y squared you could rewrite that as the quantity x over y squared here are two expressions that will help you see why you can distribute or undistribute exponents like our laws show in the first expression we have the quantity x times y squared and that's the same as x times y times x times y and the commutative property says that we can rearrange those factors like this x times x times y times y but look we can simplify that into x squared times y squared so that checks out in the second expression we have the quantity x over y squared that's the same as x over y times x over y to multiply these fractions we just multiply the tops and multiply the bottoms which gives us x times x over y times y and that simplifies to x squared over y squared so that one checks out too all right so now you know about the so called laws of exponents and there's a good chance that you'll see them explained in slightly different forms or different orders or even using different terminology in other math videos or books but the basic ideas will be the same some people like to try to memorize this list of laws and you can do that if you want to but it's an even better idea to focus on knowing how exponents really work because if you truly understand that you can actually figure out a lot of these laws for yourself and what's the best way to understand how exponents really work yep you gotta practice so be sure to do some problems with exponents on your own as always thanks for watching math antics and i'll see you next time learn more at mathantics.com so what'd you think i thought you said the guy was going to be funny