Question

Calculati media geometrica a numerelor a=radical din28 -radical din 20 totul pe radical din 3 +1 și b=radical din 12 +2 totul pe radical din 7 -radical din 5

Answer 1

[tex]a=\frac{\sqrt{28}-\sqrt{20}}{\sqrt3+1}\\ a=\frac{2\sqrt7-2\sqrt5}{\sqrt3+1}\\ a=\frac{2(\sqrt7-\sqrt5)(\sqrt3-1)}{(\sqrt3+1)(\sqrt3-1)}\\ a=\frac{2(\sqrt7-\sqrt5)(\sqrt3-1)}{3-1}\\ a=\frac{2(\sqrt7-\sqrt5)(\sqrt3-1)}{2}\\ a=(\sqrt7-\sqrt5)(\sqrt3-1)[/tex]
[tex]b=\frac{\sqrt{12}+2}{\sqrt7-\sqrt5}\\ b=\frac{2\sqrt3+2}{\sqrt7-\sqrt5}\\ b=\frac{2(\sqrt3+1)(\sqrt7+\sqrt5)}{(\sqrt7-\sqrt5)(\sqrt7+\sqrt5)}\\ b=\frac{2(\sqrt3+1)(\sqrt7+\sqrt5)}{7-5}\\ b=\frac{2(\sqrt3+1)(\sqrt7+\sqrt5)}{2}\\ b=(\sqrt3+1)(\sqrt7+\sqrt5)\\ m_g=\sqrt{ab}\\ m_g=\sqrt{(\sqrt3-1)(\sqrt3+1)(\sqrt7-\sqrt5)(\sqrt7+\sqrt5)}\\ m_g=\sqrt{(3-1)(7-5)}\\ m_g=\sqrt{2^2}\Rightarrow \boxed{\boxed{m_g=2}}[/tex]