How to write $1/ \left(1+\sqrt 3+\sqrt 5+\sqrt{15}\right)$ with a rational denominator?

Question

How to write $\frac{1}{1+\sqrt{3}+\sqrt{5}+\sqrt{15}}$ with a rational denominator?

There is an included hint: factorize the denimator

Edit: There has been some confusion on this question, the first "1" means "1 over 1+√3+√5+√15" Sorry, I can see how it could be perceived as 1/1 (1)

Answer 1

As $\sqrt{ab}=\sqrt a\cdot\sqrt b$ for $a,b\ge0,$

$$1+\sqrt3+\sqrt5+\sqrt{15}=1+\sqrt3+\sqrt5(1+\sqrt3)=(1+\sqrt3)(1+\sqrt5)$$

Now $\sqrt5+1=\dfrac{5-1}{\sqrt5-1}$

Answer 2

$$\dfrac{1}{1+\sqrt3+\sqrt5+\sqrt{15}}=\dfrac{1}{(1+\sqrt3)(1+\sqrt{5})}=\dfrac{(1-\sqrt3)(1-\sqrt{5})}{(1+\sqrt3)(1+\sqrt{5})(1-\sqrt3)(1-\sqrt{5})}=\dfrac{(1-\sqrt3)(1-\sqrt{5})}{(1-3)(1-5)}=\dfrac{(1-\sqrt3)(1-\sqrt{5})}{8}$$

Answer 3

Consider: $(1+\sqrt{3}+\sqrt{5}+\sqrt{15})(1+\sqrt{3}-(\sqrt{5}+\sqrt{15}))=-16+2(\sqrt{3}-\sqrt{65})$. So you end up with only 2 roots. Do this proces again to end up with only 1. Again and only rational numbers appear.