\(x=\frac{12}{\sqrt{3}+\sqrt{2}+\sqrt{5}}\)
=> \(x^2=\left(\frac{12}{\sqrt{3}+\sqrt{2}+\sqrt{5}}\right)^2\)
<=> \(x^2=\frac{144}{3+2+5+2\sqrt{6}+2\sqrt{10}+2\sqrt{15}}\)
<=> \(x^2=\frac{144}{2\left(5+\sqrt{6}+\sqrt{10}+\sqrt{15}\right)}\)
<=> \(x^2=\frac{144}{2\left[\sqrt{5}\left(\sqrt{5}+\sqrt{2}\right)+\sqrt{3}\left(\sqrt{5}+\sqrt{2}\right)\right]}\)
<=> \(x^2=\frac{144}{2\left(\sqrt{5}+\sqrt{2}\right)\left(\sqrt{5}+\sqrt{3}\right)}\)
<=> \(x^2=\frac{72}{\left(\sqrt{5}+\sqrt{2}\right)\left(\sqrt{5}+\sqrt{3}\right)}\)
=> \(x=\frac{6\sqrt{2}}{\sqrt{\left(\sqrt{5}+\sqrt{2}\right)\left(\sqrt{5}+\sqrt{3}\right)}}\)
