Question

​Simplify (7+√48)^1/2

Answer 1

$$\begin{gathered} {\left(7+\sqrt{48}\right)}^{1/2} \\ ={\left[7+4\sqrt{3}\right]}^{1/2} \\ ={\left[4+3+4\sqrt{3}\right]}^{1/2} \\ ={\left[{\left(2\right)}^2+{\left(\sqrt{3}\right)}^2+2\times 2\times \sqrt{3}\right]}^{1/2} \\ ={\left[{\left(2+\sqrt{3}\right)}^2\right]}^{1/2} \\ ={\left(2+\sqrt{3}\right)}^{2\times \frac{1}{2}} \\ =2+\sqrt{3} \end{gathered}$$