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given a rectangle with length a squared plus 6a minus 27 and a width a squared minus 9 right the ratio of the width of the rectangle to its length as a simplified rational expression so we want the ratio of the width we want the ratio of the width to the length to the length of the rectangle and they give us the expressions for each of these the width the expression for the width of the rectangle is a squared minus 9 so the width we do it in this pink is a squared minus 9 we want the ratio of that to the length the ratio of the width of the rectangle to its length the RET the length is given right over there it is a squared plus 6 a minus 27 they want us to simplify this and so the best way to simplify this whether we're dealing with expressions in the numerator and denominator or just numbers is we want to factor them and see if they have any common factors and if they do we might be able to cancel them out so if we factor this top expression over here what was the expression for the width this is of the form a squared minus B squared where B squared is 9 so this is going to be the same thing as a plus the square root of 9 times a minus the square root of 9 so this is a plus 3 times a minus 3 and I just recognize that from just the pattern if you ever see something a squared minus B squared it's a plus B times a minus B and you can verify that for yourself multiply this out you'll get a squared minus B squared so this is a the width can be factored into a plus 3 times a minus 3 let's see if we can do something for the denominator so here if we wanted to factor this out we have to think of two numbers that when we add them I get positive 6 and when I take the product I get negative 27 let's see if I have positive 9 and negative 3 that would work so if I this can be factored as a plus 9 and a and a minus 3 9 a 9 times a is 9 eh-eh times negative three is negative three when you add those two middle terms together you'll get six a just like that and the nine times negative three is negative 27 of course the a times a is a squared so I factored the two expressions and let's see if we can simplify it and before we simplify it because when we simplify it we lose information let's just remember what our allowable is here so we don't lose that information are there any a values here that will make this make this expression undefined well any a value that makes the denominator zero will make this undefined so a a cannot be equal to negative 9 negative 9 or 3 because if a was negative 9 or 3 then the denominator would be 0 this expression would be undefined so we have to remember this this is part of the expression we don't want to change its domain we don't want to allow things that weren't allowable to begin with so let's just remember this right over here now with that said now that we've made this constraint we can simplify it more we say look we have an a minus 3 in the denominator in the numerator and we have an a minus 3 in the denominator and we're assuming that a is not going to be equal to 3 so it's not like we're we're dividing we have a 0 over 0 so a will not be goal to 3 any other number this will be an actual number you divide the numerator and the denominator by that same value and we are left with we are left with a plus 3 over a plus 9 and the constraint here we don't want to forget the constraints are in our domain a cannot equal negative 9 or 3 it's important that we write this here because over here we lost the information that a could not be equal to 3 but in order for this to really be the same thing as this thing over here this thing over here when a was equal to 3 it wasn't defined so this in order for this to be the same thing we have to constrain the domain right over there a cannot be equal to 3 hopefully you found that useful