Question

$\sqrt{7 - 3 \sqrt{5} } - \sqrt{7 + 3 \sqrt[]{5} }$
срочнооо пожалуйста ​

Answer 1

$A=\sqrt{7-3\sqrt5}-\sqrt{7+3\sqrt5}\\\\A^2=(\; \sqrt{7-3\sqrt5}-\sqrt{7+3\sqrt5}\; )^2=\\\\=(7-3\sqrt5)-2\cdot \sqrt{(7-3\sqrt5)(7+3\sqrt5)}+(7+3\sqrt5)=\\\\=7+7-2\cdot \sqrt{7^2-(3\sqrt5)^2}=14-2\cdot \sqrt{49-9\cdot 5}=14-2\sqrt{49-45}=\\\\=14-2\sqrt4=14-2\cdot 2=10\\\\\\A^2=10\; \; \; \Rightarrow \; \; \; A=\pm \sqrt{10}\\\\Tak\; kak\; \; (7-3\sqrt5)<(7+3\sqrt5)\; ,\; to\; \; \sqrt{7-3\sqrt5}<\sqrt{7+3\sqrt5}\; \; \Rightarrow$

$\sqrt{7-3\sqrt5}-\sqrt{7+3\sqrt5}}<0\; \; \Rightarrow \; \; \; A=-\sqrt{10}\\\\\\\boxed {\; \sqrt{7-3\sqrt5}-\sqrt{7+3\sqrt5}=-\sqrt{10}\; }\\\\\\P.S.\; \; -\sqrt{10}\approx -3,1623$