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Current time:0:00Total duration:8:39

Sign of expressions challenge problems

Video transcript

let's not do some examples that the test our understanding of what happens when we multiply or divide a bunch of positive or negative numbers together and like always pause the video and see if you can work through it on your own before I do it so first question what is the sign of s to the 67th power divided by T to the ninth power divided by the expression three s times s over 40 when s is less than zero and T is greater than zero so the really interesting thing here is that s is less than zero s is negative so s is a negative if s is a negative number a negative first have s to the 67th power we just have to remind ourselves a negative number being multiplied if you have an odd number of negative numbers being multiplied together you're going to have a negative number and s to the 67th power that's 67 SS so it's literally s times s times s and you're literally going to have 67 of these s's and you're multiplying them all together so you have an odd number of s is an odd number of negative numbers that you're multiplying together so if you have an odd number of negative numbers multiplied together well that means that the product is going to be negative so this thing right over here is going to be negative now what's T to the ninth well T is a positive number they tell us that they tell us that T is greater than zero well a positive number really to any power however many you have the multiplied together with an even or odd if it's just all positive well this is going to be positive so this thing right over here is going to be positive now what's a negative divided by a positive well a negative divided by a positive is going to be negative we've already seen seen that show before so this whole thing this whole thing maybe I'll Circle it like that is going to be a negative and so that this thing doesn't get too messy actually let me let me just rewrite this so this all of this stuff that we've written over here I'll just write this as this is going to be some negative number some negative number and then we're going to divide it by three s times s over 40 now this expression right over here what is s divided by 40 going to be is that going to be positive or negative well we know that s is negative T is positive so 4 times T is going to be positive so this thing right over here is going to be positive negative divided by positive well that's going to be a negative so this so actually let me just write it this way so all of this stuff I could just replace with s divided by 40 is going to be a negative so times a negative times some other negative number now what's 3 s well s is negative so 3 s is negative so that is negative so you have a negative times a negative which is a positive so all of this business right over here is going to be positive another way you could think about it is you're going to have a negative times the negative which is positive and then divide that by positive it's just going to be positive so so all of that so this whole thing all of this business right over here is going to be a positive this is all a negative this is a positive you can have a negative divided by a positive well that's just going to be that's going to be negative so the sign of this crazy expression right over here it's going to be negative if s is less than 0 and T is greater than 0 let's keep going this is fun all right so what is the sign of P times Q over P times 4 and 2/7 when P is greater than 0 and Q is equal to 0 and the way I just dressed Q is equal to 0 should be a big hint and let me give you another hint pause the video if you really think about what I just said that Q is equal to 0 you should be able to do this in in in less than a second so why did I say that you could do this in less than a second well if you just have a bunch of numbers and then you multiply them times 0 at some point in that the whole thing is just going to be equal to 0 you know I could literally have you know a times B times C times D I could multiply a bunch of numbers and if I just knew that one of these numbers is zero then the whole product is going to be equal to 0 you know if C is equal to 0 I can multiply 8 times B times D times E and get some number and then multiply that time C is going to be anything times 0 is 0 and notice Q is equal to 0 and Q is right over here so I could take P divided by P that's going to be 1 1 times Q that's just going to be 0 which 1 times 0 0 then you have 0 times 4 and 2/7 it's just all going to be 0 and the key is you're just there's a 0 right over here at some point you're multiplying this entire product times 0 so the whole product is going or you're multiplying these numbers times 0 and so the whole product is going to be 0 so this side isn't positive or negative it's it's 0 all right there's no sign there it's neither positive nor negative all right let's do another one what is the sine of negative three-fourths times negative a to the 4th over 3 when a is less than 0 so they're telling us that a is negative so a is negative so he right over let me do let me use a more dramatic color so right over here I have a negative number to the fourth power so it's a times a times a times a and we've already said if you have an even number of negatives being multiplied by each other then that's going to be a positive you could even see it over here I mean they're all negative but a negative times the negative is going to be a positive a negative times a negative I take the third and the fourth one here that's going to be a positive and then a positive times a positive what's the whole thing the whole thing is going to be positive even though each of these are negative negative times negative or another way you could say a times a is going to be positive then times a is going to be negative but then you multiply times a again it's going to be positive again so a to the fourth power is going to be positive but then you have a negative in front of it so you're taking the negative of a positive divided by positive well so this this all of this this stuff right over here is all going to be positive but then you have this negative in front so this entire thing in front of the parenthesis or inside the parenthesis is going to be is going to be negative and then you have a negative negative 3/4 times a negative well negative times a negative the whole thing is going to be positive so what's the sign of this it's going to be positive still another I'm having too much fun all right what is the sine of X to the 59th power divided by two point three x times 4/5 when X is less than zero so X is once again a negative number so X to the 59th power that's an odd number of negatives being multiplied together so this thing right over here is going to be negative that's going to be negative right over there and this X right over here is going to be negative and so two point three times that X well that's going to be negative as well so this thing is going to be negative that's going to be negative a negative divided by a negative is a positive so all of this business is going to be a positive a positive times a positive well that's just going to be positive so all of this whole thing is going to be is going to be positive let's I can't stop this is alright what is the sine of negative x times y divided by 7/8 and here they're showing us X&Y in a number line and the key takeaways here is that X is positive x is to the right of 0 Y is negative Y is negative so let's think about this X is positive but then we're taking the negative of X right over here so X is positive but negative of X is going to be negative you bring a negative number in front of a positive number that's going to be a negative number now what's Y divided by 7/8 Y is a negative number y is a negative number I'm dividing it by a positive number so a negative divided by a positive is going to be a negative so this whole expression this whole expression right over here is going to be negative and then we have a negative all of this stuff times a negative well that's going to give us a positive so the whole thing here is going to be positive