TITLE: Rationalize $\left(\sqrt{3x+5}-\sqrt{5x+11} -\sqrt{x+9}\right)^{-1}$
QUESTION [5 upvotes]: I was trying to find if there a method similar to multiplying and dividing by the conjugate $$\frac{1}{\sqrt{3x+5}-\sqrt{5x+11} - \sqrt{x+9}},$$ but that doesn't seem to work here. Also, is there a method of multiplying by a conjugate for roots other than the square root? Such as $(ax+b)^\frac1n ± (cx+d)^\frac1m$

REPLY [4 votes]: In case you want to approach the general one, if you see the following in denominator,
$$(p^{1/n}-q^{1/m})$$
First focus on p
$$(p^{1/n}-q^{1/m})=(p^{1/n}-(q^{n/m})^{1/n})$$
multiply that by $p^{(n-1)/n}+p^{(n-2)/n}q^{n/m}+...+(q^{n/m})^{(n-1)/n}$
You get $$p-q^{n/m}=(p^m)^{1/m}-(q^n)^{1/m}$$
then again multiply this by $(p^m)^{(m-1)/m}+...(q^n)^{(m-1)/m}$
you'll get $p^m-q^n$
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