`\sqrt{15-\sqrt{216}}+\sqrt{33-12\sqrt{6}}`
`=\sqrt{15-6\sqrt{6}}+\sqrt{33-12\sqrt{6}}`
`=\sqrt{(3-\sqrt{6})^2}+\sqrt{(2\sqrt{6}-3)^2}`
`=3-\sqrt{6}+2\sqrt{6}-3`
`=\sqrt{6}`
\(\sqrt{15-\sqrt{216}}+\sqrt{33-12\sqrt{6}}\)
\(=\sqrt{15-\sqrt{6^2\cdot6}}+\sqrt{33-12\sqrt{6}}\)
\(=\sqrt{15-6\sqrt{6}}+\sqrt{33-12\sqrt{6}}\)
\(=\sqrt{\left(\sqrt{6}\right)^2-2\cdot3\cdot\sqrt{6}+3^2}+\sqrt{\left(2\sqrt{6}\right)^2-2\cdot2\sqrt{6}\cdot3+3^2}\)
\(=\sqrt{\left(\sqrt{6}-3\right)^2}+\sqrt{\left(2\sqrt{6}-3\right)^2}\)
\(=\left|\sqrt{6}-3\right|+\left|2\sqrt{6}-3\right|\)
\(=-\sqrt{6}+3+2\sqrt{6}-3\)
\(=\sqrt{6}\)
