\(\sqrt{3+\sqrt{13+\sqrt{48}}}\)
\(=\sqrt{3+\sqrt{13+4\sqrt{3}}}\)
\(=\sqrt{3+\sqrt{\left(2\sqrt{3}\right)^2+4\sqrt{3}+1}}\)
\(=\sqrt{3+\sqrt{\left(2\sqrt{3}+1\right)^2}}\)
\(=\sqrt{3+2\sqrt{3}+1}\)
\(=\sqrt{\left(\sqrt{3}+1\right)^2}=\sqrt{3}+1\)
\(\sqrt{3+\sqrt{13+\sqrt{48}}}=\sqrt{3+\sqrt{13+4\sqrt{3}}}=\sqrt{3+\sqrt{\left(2\sqrt{3}\right)^2+2.2\sqrt{3}.1+1^2}}\)
\(=\sqrt{3+\sqrt{\left(2\sqrt{3}+1\right)^2}}=\sqrt{3+\left|2\sqrt{3}+1\right|}=\sqrt{3+2\sqrt{3}+1}\)
\(=\sqrt{\left(\sqrt{3}\right)^2+2.\sqrt{3}.1+1^2}=\sqrt{\left(\sqrt{3}+1\right)^2}=\left|\sqrt{3}+1\right|=\sqrt{3}+1\)
