Question

Find the geometric mean of $\sqrt{82}-1and\sqrt{82}+1$.

Answer 1

Geometric mean of two numbers a and b: $G=\pm \sqrt{ab}$
$$\begin{gathered} \mathrm{We}\mathrm{have}: \\ a=\sqrt{82}-1\mathrm{and}b=\sqrt{82}+1 \\ \mathrm{Thus},\mathrm{we}\mathrm{get}: \\ G=\pm \sqrt{\left(\sqrt{82}-1\right)}\left(\sqrt{82}+1\right) \\ =\pm \sqrt{(82-1)} \\ =\pm \sqrt{81} \\ =\pm 9 \end{gathered}$$