[Music] hi this is laws of exponents there are nine different laws of exponents depending on which class you're in depends on which laws you'll use more frequently there are laws that you need to know more than others like anything to the zero is one which is also something that most people should find kind of simple but it's often overlooked one of my favorites is a to the negative exponent is 1 over a to the same exponent partly because I like my method of explaining it think about a rubber duck the way the rubber duck sits on the water if that is a to the negative exponent and it's sitting on the water then what that negative exponent does is it pushes it under the water and makes it a positive exponent if it's a negative exponent under the water then that negative exponent makes the rubber duck pop and make it a positive exponent where that fraction bar is the water kind of a different way to think about it but it's kind of quirky and students often don't forget it the other thing that is unusual is if you have a root students often forget the root exponent Rule and that is that the denominator is the root so if I want to find the third root then that means that that third root makes it one-third so let's go on and see if we can find some other examples first notice that I have x squared times x cubed well since my bases are the same then I'm going to add my exponents so what is two plus three well I think today 2 plus 3 is 5. so what about the second example what happens if I have an exponent on the outside of a parenthesis but I have a times on the inside if I have a times on the inside then what I do is I multiply those exponents if I have an addition or a subtraction sign inside that's like concrete and I have to foil or use distributive property twice to get those two things out that's a different video we'll get there but we're not there yet so for this one because it's a times I'm going to multiply those X exponents so this time what is 2 times 3. and what is 3 times 4. well I think this time 2 times 3 is 6 and I know that 3 times 4 is 12. now here's my negative exponent things now what about that negative sitting out in front that negative sitting out in front is just like a little puppy dog and speaking of little puppy dogs I have mine sitting right here in my lap so that little puppy dog is gonna follow you around everywhere you go so that negative sitting out in front is just gonna follow the question around but those negative exponents are going to make those num those letters move to the denominator so I'm going to end out with a negative in front and X in the numerator Y and an x squared in the denominator the next thing I need to do is to simplify those two x variables so they end up with my final answer of negative 1 over YX now I'm going to ask two things one could that denominator be written as X Y well here's a question what is 2 times 3 six what is three times two six did it matter the order in which I multiplied those numbers together no so yes I could write it as Y X or X Y it's all up to you two could I have at the very beginning of this question taken that x to the one and added 1 and negative 2 and gotten x to the negative one and then just moved it to the X but to the denominator yes there is more than one way to work a math problem and it's all up to you as long as you're following the correct mathematical steps and you get the right answer now let's look at a more complicated question so how do I do this one now I personally like to follow the order of operations so I'm going to simplify everything inside the parentheses first before I try to apply that exponent on the outside so how do I do that again I personally will take Z to the 0 is 1. then I'm going to move all my negative exponents to the bottom and deal with them and then I'm gonna go from there so I'm going to move that y to the negative 1 to the bottom and then I'm going to simplify things I find after 29 years of teaching that students tend to deal better with positive exponents than they do with negative so I tend to try to make things positive as soon as I can like I said move negative exponents to the denominator next I'm going to combine the Y's I'm going to combine the Y's to Y squared and that will get that so now I have everything inside simplified I can't go any further on the inside so next I'm going to get rid of that negative sign on the outside so how do I do that think about a fraction if I move that whole thing to the denominator then how do I divide fractions division of fraction is I flip the denominator and multiply so if I just flip that whole fraction on the inside then that gets rid of that negative exponent so in order to get rid of that negative exponent I'm flipping the whole entire fraction on the inside to get rid of that negative exponent again how do I Square everything I'm going to multiply all the exponents by 2. so that means 2 times 2 is 4 Z times Z has a first Power 1 times 2 is 2 1 times 1 is 2 so I have y to the fourth Z squared over x squared is my final answer all right next let's deal with radicals if I have just the square root of a that's understood to be a one-half so that is remember what I said the root is my denominator so that is understood to be a to the one-half [Music] in order to simplify that a has to be a perfect square in other words the square root of 4 is 2 4 is a perfect square the square root of one is one the square root of 16 is 4. they have to be perfect squares in order to simplify them what about if I have the cube root again it's understood to be the root so it's understood to be the third root so a must be a perfect Cube to be brought out of the radical so in other words 27 is a perfect cube root 27 the cube root of 27 is 3. the cube root of 8 is 2. so now let's simplify some square root now notice I put down at the bottom you can only add if things are the same number under the root and I've left that there so that you can see when to add and when to subtract and that's add and subtract okay so how can I do the first one what is the square root of 64. and what is the square root of 81 well square root of 64 is eight the square root of 81 is 9. eight minus 9 is negative 1. what about 48 48 is 4 times 12. 12 is 4 times 3. notice I have two fours so that means I can simplify this to 4 times the square root of 3. the cube root of 27 is 3. the square root of 36 is 6. three plus six is nine now even though I could not add them in their root form I can still add their answers okay make sure you understand that I can't add them in their root form but since I can find a perfect cube root for 27 and I can find a perfect square root for 36 I can add them once I simplify them the cube root of eighty-one well 81 is 3 times 27. the cube root of 27 is 3 so that's three cube roots of 3. the last one 24 is 4 times 6. 18 is 9 times 2. so 24 is 4 square roots of 6. and 24 and sorry 18 is 3 square roots of 2 but I can't simplify that anymore because the square root of 6 and the square root of 2 are not the same number under the root so that is as far as I can go with those questions all right please let me know if you have any questions thank you [Music]