loss of exponent here's the least of loss of exponent you can apply in multiplying or dividing polynomials let us start with the product law when you are multiplying expressions with the same base what you have to do is to keep the base and add their exponent for example we have here 2 cubed times 2 squared as you can see they have the same base both are two so what you have to do is to keep the base then add their exponent it will be three plus two three plus two will give us five therefore the exponent of two will become 5 and 2 raised to 5 is the same with multiplying 2 to itself 5 times so it's 2 times 2 times 2 times 2 times 2 and it will give us the result of 32 so the answer is 32. another example for example we have x raised to 5 times x raised to 4. so as you can see they have the same base therefore we have to keep the base and add their exponent 5 plus 4 will give us 9 so the answer is x raised to 9 while for quotient law since this is division the same process only that instead of adding them what you have to do is to subtract their exponent only if they have the same base the numerator and the denominator if they have the same base you have to subtract their exponent minuend is from the numerator and subtrahend is from the denominator for example 4 raised to 7 over 4 raised to 5. so as you can see they have the same base both are 4 so what we have to do is to copy the base and subtract their exponent it will be 7 minus 5. so 4 will be raised to 7 minus 5. 7 minus 5 will give us positive 2. so now we have 4 squared and when you simplify 4 squared it is the same with 4 times 4 so your answer will be positive 16 next is power of a power if an exponential expression is traced to another exponent what you have to do is to multiply their exponent for example we have here 2 cube raised to 2 so 2 has already an exponent of 3 but it is raised again to another exponent what you have to do is to multiply this two this case is different from the product law since in product law both of the expression has the base of two so here if you encounter this case what you have to do is to multiply their exponent keep the base we still have 2 and multiply 3 times 2. so 3 times 2 will give us 6 therefore it will be 2 raised to 6 and 2 raised to 6 is similar to multiplying 2 to itself 6 times so you will get so 2 times 2 is 4 times 2 is 8 times 2 is 16 times 2 is 32 and times 2 that will be 64. so the answer is positive 64. next one for power of a product if these two factors are raised to a single exponent then you have to distribute this exponent in each of this variable so it will be a raised to n and b raised to n for example we have here 3y raised to 2 or 3y squared so it means that since 3y is inside the parentheses it means both of this must be raised to 2. so therefore we will distribute this exponent to h of this 3 will have the exponent of 2 as well as the variable y will also have exponent of 2 then after that simplify 3 squared or 3 raised to 2 is just the same with 3 times 3 therefore this will be equal to 9 and then y squared just copy so the answer is 9 y squared another example 2 m cube n squared raised to 4 so 2 m cube and squared they are all inside the parenthesis that is raised to 4. therefore we have to distribute this 4 to each of these factors okay but if it has exponent what you have to do is to multiply them because we will use the law of exponent which is power of a power that if there is an exponent inside that is raised to another exponent we need to multiply them so let us start with 2 this 2 must have the exponent of 4 and then m cubed since m has an exponent it means you have to multiply its exponent to 4 so you'll have m three times four three from m and four from the exponent outside and then n squared it will be n raised to two times four so let's simplify 2 raised to 4 is the same with multiplying 2 to itself 4 times so 2 times 2 times 2 times 2 will give us positive 16 and then for m 3 times 4 of course 3 times 4 will give us 12. so we have m raised to 12. then here 2 times 4 will give us positive 8. so n raised to 8. the answer is 16 m raised to 12 and raised to 8. then for power of a quotient same with power of a product you have to distribute the exponent outside to each of these terms for example you have your x over 4 inside the parentheses and they are raised to 3. so therefore x must have the exponent of 3. also the denominator 4 must have exponent of three then after that simplify x cubed it will still be x cubed while four cubed you have to multiply four to itself so four times four times another 4 you will have positive 64. so the answer is x cubed over 64. another example 2a raised to 7 over 5b cubed raised to 2. so here 2a raised to 7 over 5b cube is inside the parentheses and they are all raised to the exponent 2. so it means each of this must be raised to 2. if the given variable has already an exponent you have to multiply them so let's start with the numerical coefficient 2 it must be raised to 2 okay the exponent is from here and then a raised to 7 raised to 2 it means what you have to do is to multiply 7 and 2 7 times 2 will give us positive 14. so a will be raised to 14. okay again we just multiply 7 and 2 then for the denominator 5b cubed raised to 2 so 5 must have an exponent of 2 and then b cube since it's already have an exponent you have to multiply it to two so three times two will give us positive six so it will be b raised to six then after that simplify two raised to 2 can still be simplified into 2 times 2 you will have positive 4 and then a raised to 14 you just have to copy it then 5 squared is the same with 5 times 5 you will have 25 and b raised to 6 you just have to copy e so the answer is 4 a raised to 14 over 25 b raised to 6. another law we have 0 zero exponent that any number raised to zero it must be equal to one as in any number even a million if it is raised to zero it is just equivalent to one for example 3y of this is raised to 0 so it means that it is equal to 1. what else for example we have 2 m cube n raised to 2 and all of these have an exponent of 0 therefore this will just be equal to 1. so again any number raised to 0 will just be equal to 1. another example what if we have x raised to 0 y squared since x raised to 0 is equal to 1 it will just be the same with 1 and then y squared okay 1 and y squared since x raised to 0 is just equivalent to 1. and we know that if the variable has a numerical coefficient of 1 we do not have to write 1. so the answer will just be y squared another law so we have here negative exponent any expression raised to a negative number you have to get its reciprocal to make the exponent positive for example 2 raised to negative 3 so the exponent is negative to make it positive you have to get the reciprocal of 2. so when you say reciprocal if it is numerator you have to write it in the denominator and vice versa so it will be since 2 is over 1 to get its reciprocal you just have to write one in the numerator and two in the denominator and once you get its reciprocal the negative exponent will automatically be positive so now we have here 2 raised to 3 1 over 2 raised to 3. and to simplify this of course you have to multiply 2 3 times to itself so it's 2 times 2 times 2 you will have positive 8 so the answer will be one over eight another what if we have a situation like this x raised to negative five and y raised to three the only negative exponent we have here is with x while the exponent of y is positive it means we will just get the reciprocal of x but not of y so y will still be in the numerator while x must be transposed in the denominator so how is that let's get the reciprocal of x so that will be 1 over x raised to 5 while y will still be in the numerator so multiplying 1 and y cube it will just be equal to y cube and the numerator and then x raised to 5 in the denominator that means those with positive exponent will still be in the same position and those with negative exponent you have to um get its reciprocal so now we have y cubed over x raised to five second case we have one over b raised to negative n so the variable with the negative exponent is in the denominator vice versa to make it positive still you have to get its reciprocal so since it's 1 over b it will become b over 1 or it will just be equal to b raised to n because 1 as the denominator we are not writing it anymore for example 1 over 3 raised to negative 2. so this 3 raised to negative 2 we just have to write it in the numerator while 1 must be in the denominator or in short this will just be equal to 3 raised to 2 because we are not writing the denominator 1 anymore okay we do not need this so we we may just write 3 raised to 2 and 3 raised to 2 is the same with multiplying 3 to itself 3 times 3 that will be equal to positive 9. another example what if we have this x raised to negative 2 over x raised to negative 3 both of their exponents are negative so to get the reciprocal it means you just have to exchange their position it means y will be in the numerator and x will be in the denominator so after you do that their exponent will be positive y from the denominator will be in the numerator and x from the numerator will be written in the denominator [Music] you