\(\sqrt{7+\sqrt{24}}=\sqrt{7+2\sqrt{6}}=\sqrt{\left(\sqrt{6}\right)^2+2.\sqrt{6}.1+1^2}\)
\(=\sqrt{\left(\sqrt{6}+1\right)^2}=\left|\sqrt{6}+1\right|=\sqrt{6}+1\)
\(\sqrt{2-\sqrt{3}}=\sqrt{\dfrac{4-2\sqrt{3}}{2}}=\sqrt{\dfrac{\left(\sqrt{3}\right)^2-2.\sqrt{3}.1+1^2}{2}}\)
\(=\sqrt{\dfrac{\left(\sqrt{3}-1\right)^2}{2}}=\dfrac{\left|\sqrt{3}-1\right|}{\sqrt{2}}=\dfrac{\sqrt{3}-1}{\sqrt{2}}\)
\(\sqrt{5+2\sqrt{6}}-\sqrt{5-2\sqrt{6}}\)
\(=\sqrt{\left(\sqrt{3}\right)^2+2.\sqrt{3}.\sqrt{2}+\left(\sqrt{2}\right)^2}-\sqrt{\left(\sqrt{3}\right)^2-2.\sqrt{3}.\sqrt{2}+\left(\sqrt{2}\right)^2}\)
\(=\sqrt{\left(\sqrt{3}+\sqrt{2}\right)^2}-\sqrt{\left(\sqrt{3}-\sqrt{2}\right)^2}=\left|\sqrt{3}+\sqrt{2}\right|-\left|\sqrt{3}-\sqrt{2}\right|\)
\(=\sqrt{3}+\sqrt{2}-\sqrt{3}+\sqrt{2}=2\sqrt{2}\)
