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Multiplying and Dividing Rational Expressions - Concept

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Teacher/Instructor Carl Horowitz
Carl Horowitz

University of Michigan
Runs his own tutoring company

Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!

Dividing rational expressions is basically two simplifying problems put together. When dividing rationals, we factor both numerators and denominators and identify equivalents of one to cancel. After identifying these equivalents, we take the reciprocal of the second fraction and divide. Multiplying rational expressions is the same as dividing rationals, except that we do not take the reciprocal of the second fraction.

Multiplying and dividing rational expressions is very similar to multiplying and dividing fractions okay if we have 3, 4's times 8, 7's you already know that we can cancel like terms if we have them so 4 and 8 share a factor 4 so we know that we can turn this down to 1, turn this to a 2 and then just multiplying across this ends up being 6 over 7. Okay we didn't have to simplify this we could end up getting 3 times 8, 24 over 4 times 7, 28 and then simplify it but in general if we can simplify it before hand life becomes a lot easier.
Okay with rational expressions it's no different okay so what we have here is a rational expression and we're just going to cancel like terms for 4, so we could have cancel the 4 over 4 here to begin with or we can cancel the 4 and the 8 it doesn't really matter because it's all going to be multiplied in the end so I'm going to cancel the 4 and 4 here okay we then have a y in the bottom and a y squared in the top so I'm going to go ahead and cancel one of those y's, x cubed in the top x the fifth in the bottom so this is 3x's and this is 5x's we can actually go ahead and cancel 3 of those so then this ends up being x squared. Once we've simplified everything up we can just multiply across so our numerator now just has an 8y and our denominator has just a x squared okay so by cancelling things just like we did when we were dealing with fractions we're able to simplify this up fairly easily okay.
Division just like with fractions is basically the same process where you take a division and then you flip your divisor and multiply okay so with the fraction what we're used to is 5 sixth divided by 10 thirds just becomes 5 sixth times 3 over 10. This is the same thing we did back there where we can cancel anything we have in common so 3 and 6 are both divisible by 3 cancel out the 3, 5 and 10 both divisible by 5 cancel the 5 so we end up with one fourth.
Okay when dealing with rational expressions, this one is pretty ugly but we can still do the exact same thing so our first term always stays the same so we end up with 5 x the forth, y squared over 16 x squared, y times and then our divisor our second time flips over so this ends up being 60x cubed y squared divided by 25 x squared y. Okay so we can either cancel across our fractions or within a fraction it doesn't really matter okay so when I'm going to end up doing is I have a y in the bottom another y over here and a y squared at top so those 3 things can all cancel to get nothing okay we then have a x squared and an x squared in the bottom and an x to the forth in the top those can cancel, 5 and 25 cancel down to 5 and 16 and 60 both divisible by 4 so that becomes a 4 and that becomes, believe it's 15, 15 and 5 once again can cancel leaving us with 3 okay so just by going through I'm looking for common terms so we're able to simplify up what we're actually multiplying. Then just multiplying across seeing what we're left with that's not crossed out, everything in this numerator cancels out, I have a 3x cubed y squared and then we have a 4 and then that's it in the denominator and then just checking to make sure that I didn't miss anything 3x square 3x cube y squared over 4 can't cancel so what we're able to do is by flipping our devisor multiplying cancelling anything that can cancel we're able to simplify this up.
Okay, so multiplying rational expressions pretty much as the same as multiplying fractions dividing rational expressions pretty much the same as dividing fractions all you have to do is flip it over and then turn it right back into the multiplication problem.

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