Question

2.sa se aduca la forma cea mai simpla
a.√(6+2√5)
b.√(17+√288)
c.√(28-16√3)

Answer 1

$\sqrt{6+ 2\sqrt{5} } = \sqrt{6+ \sqrt{20} } \\ \sqrt{A+ \sqrt{B} }= \sqrt{ \frac{A+C}{2}} +\sqrt{ \frac{A-C}{2}} \\ C= \sqrt{ A^{2} -B} \\ \\ C= \sqrt{6^{2}-20} = \sqrt{36-20} = \sqrt{16} =4 \\ C=4 \\ \sqrt{6+ \sqrt{20} }= \sqrt{ \frac{6+4}{2} } + \sqrt{ \frac{6-4}{2} } = \sqrt{ \frac{10}{2} +1} =\sqrt{ {6}}$
b.$\sqrt{17+ \sqrt{288} } \\ C= \sqrt{ A^{2}-B} = \sqrt{17^{2}-288 } = \sqrt{289-288} =1 \\ \sqrt{A+ \sqrt{B} } = \sqrt{ \frac{A+C}{2} } + \sqrt{ \frac{A-C}{2} }=\sqrt{ \frac{17+1}{2} } + \sqrt{ \frac{17-1}{2} }=3+2 \sqrt{2}$
c.$\sqrt{28-16 \sqrt{3 }} = \sqrt{28- \sqrt{768 }} \\ C= \sqrt{A^{2} -B} = \sqrt{784-768} \\ = \sqrt{16} =4 \\ \sqrt{28-16 \sqrt{3 }} = \sqrt{ \frac{28+4}{2} } -\sqrt{ \frac{28-4}{2} } =4-2 \sqrt{3}$