Simplify by rationalizing the denominator.

Question

I have the problem:

Write $$\frac{4 \sqrt{2} + 6 \sqrt{3} + 10 \sqrt{5}}{(\sqrt{2} + \sqrt{3} + \sqrt{5})^2}$$in simplest form.

I have tried simplifying by doing this: $$\frac{4(\sqrt2+\sqrt3+\sqrt5)+2\sqrt3+6\sqrt5}{(\sqrt2+\sqrt3+\sqrt5)^2}$$ but that hasn't seemed to get me anywhere. Could someone show me the steps to the solution.

Answer 1

Hint

$$\dfrac{2(a^3+b^3+c^3)}{(a+b+c)^2}=?$$

Here $a=\sqrt2,b=\sqrt3,c=\sqrt5$

Observe that $a^2+b^2=c^2$

Answer 2

As your question is "rationalise the denominator" one can expect that this will involve rationalising the denominator. For 3 terms you do the same as for 2 terms but need to group a pair and do it twice. E.g. grouping $\sqrt 2$ and $\sqrt 3$ will leave yoy with a $\sqrt 6 term that will need to be taken care of in a second step.