Simplify
$$\dfrac{2\sqrt{21}-\sqrt{35}+5\sqrt{15}-16}{\sqrt7+2\sqrt5-\sqrt3}$$Final solution should have rational denominators.
Suppose the solution is $X$, I have tried to make up an equation for $X^2$
$$X^2 = \frac{200}{-2\sqrt{15}+2\sqrt{35}-\sqrt{21}+10}$$
My idea is that solving for $X^2$, if can be simply done, can easily give us $X$.
Please help!
Best Answer
Hint: we are generalising the method of rationalising a denominator of two terms. For instance, if our denominator was $\sqrt{a}-b$ then we would multiply (numerator and denominator) by $\sqrt{a}+b$ to get $a-b^2$. That is, we are using "difference of two squares".
Observe that $[(\sqrt{7}+2\sqrt{5})-\sqrt{3}][(\sqrt{7}+2\sqrt{5})+\sqrt{3}]=(\sqrt{7}+2\sqrt{5})^2-(\sqrt{3})^2=7+4\sqrt{35}+20-3$
Now what can we do?