Simplifying Rational Expressions

in this lesson we're going to focus on simplifying rational expressions so let's start with our first example to the fifth over 49 squared go ahead and simplify this expression we can reduce 35 and 49 by seven 35 divided by seven is five 49 divided by 7 is 7. now whenever you're dividing by common base you're allowed to subtract the exponents 5 minus 2 is 3. so the final answer for this problem is 5x cubed divided by seven now you can also see it this way let's start from the beginning 35 is seven times five to the fifth is basically five variables multiplied to each other 49 is 7 times 7 and squared is just times in this lesson we're focusing on cancelling and simplifying we can cancel 7 and we can cancel two variables so we have five on top three variables remaining and seven on the bottom which is why this is the final answer what about this example 4x squared plus 8x divided by 3x plus 6. how can we simplify this expression in this example we need to factor out the gcf the greatest common factor of the two terms on top is 4x 4x squared divided by 4x is 8x divided by 4x is positive 2. now on the bottom we could take out the gcf which is stream 3x divided by 3 is 6 divided by 3 is 2. now notice that we can cancel plus 2. so therefore the final answer is 4x divided by 3. here's another example squared minus 16 divided by squared plus 9x plus 20. so what do you think we need to do in this problem your goal is to factor completely as possible and cancel we know how to factor squared minus 16 using the difference of squares technique it's going to be plus 4 and minus 4. you just got to take the square root of squared which is and the square root of 16 which is 4. now in this example below we have trinomial with leading coefficient of one to factor it all we need to do is find two numbers that multiply to 20 but add to nine this is clearly four and five so it's going to be plus four times plus five and we can clearly see that plus four will cancel so the final answer is minus four divided by plus five and that's how you could simplify rational expressions but there's one more example i'm going to go over you might see this in your homework five minus over minus five these two look very similar but they don't cancel immediately whenever you see situation like this factor out negative one it could be from the top or the bottom i'm going to take it from the top if take away negative one from negative it will change to positive and write the first before you write the five so reverse the order if take out negative one from positive five it will change sign it will become negative five so notice that we can cancel minus five at this point so the final answer for this is negative one now based on the examples that we've covered so far go ahead and work on these practice problems start with this one simplify the expression 72 to the 8 to the 7 divided by to the fifth to the fourth for each of these feel free to pause the video and work on it 72 is basically eight times nine to the eighth is to the five times to the three because five plus three is eight to the seven is to the fourth times to the three sixty four is eight times eight now we could cancel an eight we could cancel to the fifth and we could cancel to the fourth so the final answer is what we have left over on top we have nine cubed to the third and on the bottom we just have an 8. so that's the answer for that problem now let's move on to the next example so try this one 5x squared minus 15x divided by 8x minus 24. go ahead and simplify the rational expression so in the denominator we can take out the gcf the greatest common factor which is 8. 8x divided by 8 is negative 24 divided by 8 is negative 3. in the numerator we can take out 5x 5x squared divided by 5x is and negative 15x divided by 5x is negative 3. so we could cancel the common term minus three so the final answer is five divided by eight so that's it for this problem here's another one forty two minus 6x divided by 3x minus 21. so once again let's take out the greatest common factor in the numerator we can take out 6. 42 divided by 6 is 7. negative 6x divided by 6 is negative in the denominator we can take out 3. 3x divided by 3 is negative 21 divided by 3 is negative and notice that we have this situation so we need to factor out negative one let's take out negative one from seven minus and reverse the letters or numbers and letters so negative becomes positive if we do so and positive seven will change into negative seven and now at this point we could cancel minus seven and so we're left with negative six divided by three which we know it's negative two and that is the answer for this one try this problem squared minus eight plus fifteen divided by two squared minus eighteen so let's factor the trinomial which has leading coefficient of one so what two numbers multiply to 15 but add to the middle coefficient negative eight we know that five times three multiplies to fifteen but adds up to positive eight so we need to make it negative five and negative three because they still multiply to positive 15 but this time they add up to negative eight so it's going to be minus five times minus three now what about the bottom what should we do it looks like difference of squares situation but the square root of 18 that's not going to be nice integer however we can take out the gcf which is 2. 2x squared divided by 2 is squared and negative 18 divided by 2 is negative 9. now we can use the difference of squares technique the square root of squared is and the square root of nine is three one is going to be positive and the other will be negative so therefore the final answer is minus five divided by two times plus 3. and so that's it there's another one that you could try 2x squared minus 5x minus 3 divided by four squared minus one go ahead and take minute and work on that so how can we factor two squared minus five minus three this time we have trinomial where the leading coefficient is not one so we need to multiply two by negative three two times negative three is negative six and we need two numbers that multiply to negative six but add to negative five so that's going to be negative 6 and positive 1. so now replace negative 5x with negative 6x plus 1x as you can see that's terrible three let's do that again these two add up to negative 5x now we need to factor by grouping so in the first two terms take out the gcf which is going to be 2x and so we're going to be left with minus and for the last two terms if there's nothing to take out take out one so we're going to get minus three again now let's factor out minus three since we have common term and then we're going to get 2x plus 1. and so that's how you can factor the numerator now to factor the denominator we have the difference of perfect squares situation the square root of 4x squared is two and the square root of one is one and so it's just gonna be plus or minus which means we can cancel two plus one so the final answer is minus 3 divided by 2x minus 1. so now you know how to simplify rational expressions factor take out the gcf and cancel and that's basically it so thanks for watching you