Question

Media geometrică a numerelor: a = (2 - radical (3))(radical (5) + 2) * si = (2 + radical (3))(radical(5) - 2) este egală cu​

Answer 1

folosim formula de calcul prescurtat:

$\boxed {(a - b)(a + b) = {a}^{2} - {b}^{2}}$

calculăm media geometrică:

$m_{g} = \sqrt{ab} = \sqrt{(2-\sqrt{3})(\sqrt{5}+2)(2+\sqrt{3})(\sqrt{5}-2)} =$

$= \sqrt{\Big[(2+\sqrt{3})(2-\sqrt{3})\Big]\Big[(\sqrt{5}+2)(\sqrt{5}-2)\Big]}$

$= \sqrt{\Big(2^{2} - \sqrt{3}^{2}\Big) \Big(\sqrt{5}^{2} - 2^{2} \Big)} = \sqrt{\Big(4 - 3\Big) \Big(5 - 4\Big)}$

$= \sqrt{1 \cdot 1} = \sqrt{1} = \bf 1$