\(T=\left(2\sqrt{3}+1\right)\left(3\sqrt{2}-1\right)\sqrt{13-4\sqrt{3}}.\sqrt{19+6\sqrt{6}}\)
\(T=\left(2\sqrt{3}+1\right)\left(3\sqrt{2}-1\right)\sqrt{\left(2\sqrt{3}-1\right)^2}.\sqrt{\left(3\sqrt{2}+1\right)^2}\)
\(T=\left(2\sqrt{3}+1\right)\left(3\sqrt{2}-1\right)\left|2\sqrt{3}-1\right|.\left|3\sqrt{2}+1\right|\)
\(T=\left(2\sqrt{3}+1\right)\left(3\sqrt{2}-1\right)\left(2\sqrt{3}-1\right)\left(3\sqrt{2}+1\right)\)
\(T=\left(2\sqrt{3}+1\right)\left(2\sqrt{3}-1\right)\left(3\sqrt{2}-1\right)\left(3\sqrt{2}+1\right)\)
\(T=11\cdot17\)
\(T=187\)