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# Intro to rationalizing the denominator

## Video transcript

in this video we're going to learn how to rationalize the denominator rationalize rationalize the denominator what we mean by that is let's say we have a fraction that has a non rational denominator the simplest one I can think of is one over the square root of two so to rationalize this denominator we're going to just rear entrance um way that does not have an irrational number in the denominator now the first question you might ask is Sal why do we care why must we rationalize denominators and you don't have to rationalize them but I think the reason why this is in many algebra classes and why many teachers want you to is it gets the numbers into a common format and I also think that I've been told that back in the day before we had calculators that made some forms of computation people found it easier to have a rational number in the denominator I don't know if that's true or not and then the other reason is just for aesthetics some people say I don't know I don't like saying what one square root of tooth is I don't even know you know I want to know how big the pie is so I want a denominator to be a rational number so with that said let's learn how to rationalize it so the simple way if you just have a simple rat irrational number the denominator just like that you can just multiply the numerator and the denominator by that irrational number over that irrational number now this is clearly just one anything over anything or anything over that same thing is going to be one so we're not fundamentally changing the number we're just changing how we represent it so what's this going to be equal to the numerator is going to be 1 times the square root of 2 which is the square root of 2 the denominator is going to be the square root of 2 times the square root of 2 well the square root of 2 times the square root of 2 is 2 that is 2 right that by definition this squared must be equal to 2 and we are squaring it we're multiplying it by itself so that is equal to 2 we have rationalized the denominator we haven't gotten rid of the radical sign but we've brought it to the numerator and now in the denominator we have a rational rational number and you could say hey now I have square root of 2 have's it's easier to say even so maybe that's another justification for rationalizing this denominator let's do a couple more examples let's say I had seven over the square root of fifteen so the first thing I'd want to do is just simplify this radical right here let's see square root of 15 15 is 3 times 5 neither of those are perfect squares so actually this is about as simple as I'm going to get so like just like we did here let's multiply this times the square root of 15 over the square root of 15 and so this is going to be equal to 7 times the square root of 15 just multiply the numerators over square root of 15 times the square root of 15 that's 15 so once again we have rationalized the denominator this is now a rational number we essentially got the radical up on the top where we got the irrational number up on the numerator we haven't changed the number we just changed how we are representing it now let's take it up one more level what happens if we have something like 12 over 2 minus the square root of 5 so in this situation we'll actually have a binomial in the denominator and this binomial contains an irrational number I can't do the trick here if I multiply this by square root of 5 over square root of 5 I'm still going to have a rational denominator let me just show you just to show you it won't work if I multiply this square root of 5 over square root of 5 the numerator is going to be 12 times the square root of 5 the denominator we have to distribute this it's going to be 2 times the square root of 5 minus the square root of 5 times square to 5 which is 5 so you see in this situation it didn't help us because the square root of 5 although this day part became rational became a 5 this part became irrational 2 times the square root of 5 so this is not what you want to do where you have a binomial that contains an irrational number in the denominator what you do here is use our skills when it comes to difference of squares so let's just take a little side here we learned a long time ago well not that long ago that if you had a look if you had 2 - is the square root of five and you multiply that by two plus the square root of five what will this get you now you might remember and if you this you don't recognize this immediately this is the exact same pattern as a minus B times a plus B right a minus B times a plus B which we've seen several videos ago is a squared minus B squared a little bit of review this is a times a which is a squared a times B which is a B minus B times a which is minus a B and then negative B times positive B negative B squared these cancel out you're just left with a squared minus B squared so two minus the square root of five times two plus the square root of five is going to be equal to 2 squared 2 squared which is 4 let me write it that way it's going to be equal to 2 squared minus square root of 5 squared which is just 5 so this would just be equal to 4 minus 5 or negative 1 so if you multiply it by if you if you take advantage of the difference of squares of binomials or that the factoring difference of squares however you want to view it then you can rationalize this denominator so let's do that so if I have let me rewrite the problem 12 over 2 minus the square root of 5 in this situation I just multiply the numerator and the denominator by 2 plus the square root of 5 over 2 plus the square root of 5 once again I'm just multiplying the number by 1 so I'm not changing the fundamental number I'm just changing how we represent it so the numerator is going to become 12 times 2 which is 24 plus 12 times the square root of 5 plus 12 times the square root of 5 all of that over the this we once again this is going to be this is like a factored difference of squares so this is going to be 2 this is going to be equal to 2 squared which is it's going to be exactly equal to that which is 4 minus 1 or we could just read or sorry 4 minus 5 right it's 2 squared minus square root of 5 I've squared so it's four minus five or we could just write that as minus one or negative one we could put a 1 there and put a negative sign out in front and then you know no point even putting a 1 in the denominator we could just say that this is equal to negative 24 minus 12 square roots of 5 so in this case it kind of did simplify it well wasn't just for the sake of rationalizing it actually made it look a little bit better and you know I don't know if I mentioned in the beginning this is good because you know it's not obvious if you and I are both trying to build a rocket and you get this as your answer and I get this as my answer this isn't obvious at least to me just by looking at that they're the same number but if we agree to always rationalize our denominators or like oh great we got the same number now we're ready to send our rocket to Mars let's do one more of this one more of these right here let's do one more let's say let's say I have let me do let's say let's do one with variables in it so let's say we have 5y over 2 times the square root of Y minus 5 so we're going to do this exact same process we have a binomial with the irrational denominator it might be a rational time we don't know what Y is but Y can take on any value so it's points it's going to be irrational so we really just don't want a radical in the denominator so what is this going to be equal to let's just multiply the numerator denominator by 2 times this 2 square roots of y plus 5 over 2 square roots of y plus 5 this is just 1 we are not changing the number we're just multiplying it by 1 so let's start with the denominator what is the denominator going to be equal to the denominator is going to be equal to this squared is 2 once again just a difference of squared so it's going to be 2 times the square root of Y squared minus 5 squared right if you factor this you would get 2 square roots of y plus 5 times 2 squared so y minus 5 this is a difference of squares and then our numerator is 5 y times 2 square roots of Y so would be 10/10 and this is why to the first power this is why to the half power we could write Y square roots of Y 10 y square roots of Y or we could write this as Y to the 3 halves power or one and a half power however you want to view it and then finally 5y times five is plus 25y and we can simplify this further so what is our denominator going to be equal to we're going to have 2 squared which is 4 square root of Y squared is y right 4y and then minus 25 minus 25 and our numerator our numerator over here is we could even write this we could keep it exactly the way we've written it here we could factor out a y there's all sorts of things we could do it but it just to keep just to keep things simple we could just leave that as 10 let me just write it different I could write that as this is y to the first this is y to the one-half power I could write that as even in Y to the three halves if I want I could write that as Y to the one and a half if I want or I could write that as 10 y times the square root of Y all of those are equivalent plus 25 plus 25 y anyway hopefully you found this rationalizing the denominator interesting