Express `sqrt(20) - (45/sqrt(5))` in the form `k sqrt(a).`

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In general, it is in bad form to have a radical in the denominator of a fraction.  When you come across this, you must do what's called "rationalizing the denominator".  The best way to ensure you get rid of the radical is to multiply your fraction by by a new fraction that has the radical in both the numerator and the denominator (and therefore has a value of 1).  It would look like this:


It is very important to multiply the top and bottom of you original fraction by the same thing so that you don't change the value of the fraction!

Next you multiply your fractions together and get:


Now we are looking at the expression


You are only allowed to add or subtract radicals that are the same.  So we want to simplify the square root of 20 in the hopes that it will contain the square root of 5:


Now we can simplify our expression.  We only add or subtract the coefficients in front of the radical (and keep the radical the same):


So k=-7 and a=5

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Probably `k` and `a` should be integers:)  To find them, first consider `sqrt(20)` and `45/sqrt(5)` separately.

`20 = 4*5 = 2^2*5,` therefore `sqrt(20) = 2*sqrt(5).`

`45 = 9*5` and `5/sqrt(5) = sqrt(5),` so `45/sqrt(5) = 9*(5/sqrt(5)) = 9*sqrt(5).`

Thus the difference is `2sqrt(5)-9sqrt(5)=-7sqrt(5),` and we found that `a=5` and `k=-7.` This is the answer.

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