Question

A= radical din 7 - radical din 5
B= radical din 5 - radical din 3
a) Demonstrati ca A la puterea -1 > la puterea -1
b) Aratati ca 1/a - 1/b = a+b/2

Answer 1

[tex]a=\sqrt7-\sqrt5\\ b=\sqrt5-\sqrt3\\\\ a)A^{-1}\ \textgreater \ B^{-1}\\ \frac{1}{\sqrt7-\sqrt5}\ \textgreater \ \frac{1}{\sqrt5-\sqrt3}\\ \frac{\sqrt7+\sqrt5}{2}\ \textgreater \ \frac{\sqrt5+\sqrt3}{2}|\cdot 2\\ \sqrt7+\sqrt5\ \textgreater \ \sqrt5+\sqrt3\\ \sqrt7\ \textgreater \ \sqrt3 |()^2\\ 7\ \textgreater \ 3 (A)\\ \\ b)\frac{1}{a}-\frac{1}{b}=\frac{a+b}{2}\\ \frac{1}{\sqrt7-\sqrt5}-\frac{1}{\sqrt5-\sqrt3}=\frac{\sqrt7-\sqrt5+\sqrt5-\sqrt3}{2}\\ \frac{\sqrt7+\sqrt5}{2}-\frac{\sqrt5+\sqrt3}{2}=\frac{\sqrt7-\sqrt3}{2}\\ \frac{\sqrt7+\sqrt5-\sqrt5-\sqrt3}{2}=\frac{\sqrt7-\sqrt3}{2}\\ [/tex]
$\frac{\sqrt7-\sqrt3}{2}=\frac{\sqrt7-\sqrt3}{2}(A)$