in this video we're going to focus on simplifying exponents but let's go over some basic properties what is X 4th multipli X 5th what is that equal to whenever you multiply by a common base you need to add the exponents 4 + 5 is 9 x to the 4th is basically multiplying four x variables together and X the 5th is equivalent to multiplying 5 X variables together so in total you're multiplying 9 x variables together and so that's why it's equal to x to the 9th now what about division let's say if you divide by a common base what should you do with the exponents you need to subtract x 7 / x^ 3 we need to subtract 7 by 3 and that's going to be X 4 x 7 is equivalent to 7 x variables multiply to each other and X Cub is simply x * x * X so we can cancel three x variables which leave behind four x variables on top and so that's the answer here's another one what is X 3r raised to the 4th power what should we do whenever we raise one exponent to another in a situation like this you need to multiply it's going to be 3 * 4 which is 12 so x 3 rais 4th power means that you're multiplying 4 x Cub values and each X cube is basically three x variables multiply to each other so if you count all the X variables what we have is a total of of 12 and so that's why it's equal to x 12 now what about X raised to the 0 power what is that equal to anything raised to the 0 power is one so 4 to the 0 is 1 now what about X raised to the -3 what should we do if we have negative exponents if you have a negative exponent all you could do is take the variable move it to the bottom when you do that the -3 will become positive3 and so to 1x Cub likewise let's say if you have 1 / X raised to4 you can move the X variable to the top and the exponent will change sign so it's going to be X pos4 over 1 or simply X raised 4th power now here's a question for you what is the value of -32 what do you think that's equal to now what about -32 within a parentheses -32 is basically -3 * -3 we have two of them so this is equal to POS 9 now what we have here is 1 netive and two 3es this two does not apply to the negative sign so we only have one negative sign so this is negative and then 3 * 3 is 9 so the whole answer is9 so just keep in mind the difference between the two you can confirm this in the calculator if you type in -32 just the way you see it like this you should get ne9 but if you type it like this in your calculator it should give you positive 9 go ahead and simplify these problems simplify the following Expressions so let's say this is the first one here's the next One X Cub raised to the5 and also X 7th power / x 12 go ahead and work on these examples so when multiplying by when multiplying common bases you need to add the exponents 4 + 9 is5 and X raised to5 is equivalent to 1 x 5th so if you get any negative exponents make it positive now here we have one exponent raised to another so we need to multiply 3 * 5 is-15 X15 is 1/ X pos5 so this is the final answer for that problem now for the last one we need to subtract we need to take the top number and subtract it by the bottom number 7 - 12 is5 and this is the same as 1 over X to 5 now let's talk about that example X 7th means that we have seven X variables X the 12 means that we have a total of 12 x variables multiplied to each other so we can cancel all seven X variables on top and seven on a bottom and notice that we have five x variables left on the bottom and that's y it's 1/ x 5 what is the value of 3x^2 raised to the 3 power go ahead and simplify this expression so three is the same as 3 to the 1 whenever you raise one exponent to another you need to multiply but we have two exponents here so we need to distribute the three to the one and to the 2 so 3 * 1 is 3 and 2 * 3 is 6 so this is 3 3r * x 6 so what's 3 3r 3 3r is 3 * 3 * 3 3 * 3 is 9 and 9 * 3 is 27 so the final answer is 27 * X raised to six power try this one -2 okay that doesn't look like a two let's do that again -2X Cub y 4th raised to the second power feel free to pause the video and work on that example so this is raised to the first Power and if we multiply by two we're going to have -2 to the second power and then 3 * 2 is 6 so we're going to have X to 6 and 4 * 2 is 8 so this is going to be y 8 -2 2 that's basically -2 * -2 that's equal to pos4 so the answer is 4 x to 6 y raised to the 8th power what is the value of 5x Cub multiplied 4X 7th power so what should we do in this example so first we need to multiply 5 * 4 5 * 4 is 20 and then we need to multiply X Cub * X 7th in which case we need to add 3 + 7 is 10 so this is simply 20 x 10th so try this example 7 x 6 * 5 x 4 and also 4X cub y^2 * 7 X4 y 3r so in the first example we need to multiply 7 by 5 7 * 5 is 35 and then we got to multiply x 6 * x 4 6 + 4 is 10 so it's 35 x 10 now the next example we got to multiply 4 by 7 which is 28 and then X Cub * X4 so we got to add 3 +4 which is1 and then finally we got to multiply y^2 * y Cub 2 + 3 is 5 so it's y the 5th power now we need to get rid of the negative exponent so let's move the X variable to the bottom so it's 28 y 5 / X and so that's the answer what is -2 x 2 y raised to 0 power what is that equal to anything raised to 0 power keep in mind it's one now what about -4x 0 power what is that equal to you need to realize that this is -4 * X raised to0 power this zero only applies to X so this portion highlighted in red that's equal to 1 so it's -4 * 1 and that's so anything that's affected by the zero exponent will become a one anything outside of that is unaffected try this problem what is 24 x 7th y 3r / 8 x 4 y^ the -2 so in this case we need to divide 24 / 8 is 3 x to 7th / X 4th is X Cub 7 - 4 is 3 and for the Y variables we need to subtract 3 - -2 is the same as 3 + 12 so this becomes y the 15th and so that's the answer what is 35 x Cub y 5 / 63 X4 Y 7th squared what is that equal to so we could distribute the two or we could simplify first we could divide first or distribute the exponent let's divide first 35 is basically 5 * 7 and 63 is 9 * 7 now x 3r / X4 that's going to be 3 - -4 which is 3 + 4 and that's 7 so we're going to have x 7 on top and then we have 5 - 7 which is -2 it's -2 on top but if we bring it to the bottom it's going to be positive2 on the bottom so let's go ahead and do that and this is still raised to the second power now the last thing we could do is cancel the seven let's get rid of this stuff so we have 5 x 7th / 9 y^2 raised the 2 power 5^ 2 is 25 and X 7th raised the 2 power is going to be x 14 7 * 2 is 14 9 2 is 81 and 2 * 2 is 4 so this should be the final answer