`\sqrt{2+\sqrt{3}}-\sqrt{2-\sqrt{3}}`
`={\sqrt2(\sqrt{2+\sqrt3}-\sqrt{2-\sqrt3})}/{\sqrt2}`
`={\sqrt2\sqrt{2+\sqrt3}-\sqrt2\sqrt{2-\sqrt3}}/{\sqrt2}`
`={\sqrt{4+2\sqrt3}-\sqrt{4-2\sqrt3}}/{\sqrt2}`
`={\sqrt{(\sqrt3+1)^2}-\sqrt{(\sqrt3-1)^2}}/{\sqrt2}`
`=frac{|\sqrt{3}+1|-|\sqrt{3}-1|}{\sqrt{2}}`
`=frac{(\sqrt{3}+1)-(\sqrt{3}-1)}{\sqrt{2}}` (do `\sqrt{3}+1>0;\sqrt{3}-1>0)`
`={\sqrt3+1-\sqrt3+1}/{\sqrt2}`
`=(2)/(\sqrt{2})`
`=\sqrt{2}`
