Question

Ajutatima va rog la ex 31 pag 35 M. clasa 8
Dau corona!!!​

Answer 1

Explicație pas cu pas:

a)

$\dfrac{ \sqrt{2 + \sqrt{3} } }{ \sqrt{2 - \sqrt{3} } } = \dfrac{ \sqrt{(2 + \sqrt{3})(2 + \sqrt{3} ) } }{ \sqrt{(2 - \sqrt{3})(2 + \sqrt{3}) } } = \\ = \dfrac{ \sqrt{ {(2 + \sqrt{3} )}^{2} } }{ \sqrt{4 - 3} } = \dfrac{ |2 + \sqrt{3} | }{ \sqrt{1} } = 2 + \sqrt{3}$

b)

$\dfrac{ \sqrt{2} }{ \sqrt{5} - \sqrt{7 - \sqrt{2} } }=$

$= \dfrac{ \sqrt{2}( \sqrt{5} + \sqrt{7 - \sqrt{2} })}{(\sqrt{5} - \sqrt{7 - \sqrt{2} } )(\sqrt{5} + \sqrt{7 - \sqrt{2} } )}$

$= \dfrac{ \sqrt{2}( \sqrt{5} + \sqrt{7 - \sqrt{2} })}{5 - 7 + \sqrt{2}}$

$= \dfrac{ \sqrt{2}( \sqrt{5} + \sqrt{7 - \sqrt{2} })}{\sqrt{2} - 2}$

$= \dfrac{ \sqrt{2}( \sqrt{5} + \sqrt{7 - \sqrt{2} })( \sqrt{2} + 2)}{(\sqrt{2} - 2)( \sqrt{2} + 2 )}$

$= \dfrac{( \sqrt{5} + \sqrt{7 - \sqrt{2} })(2 + 2 \sqrt{2} )}{- 2}$

$= - ( \sqrt{5} + \sqrt{7 - \sqrt{2} })(1 + \sqrt{2} )$

c)

$\dfrac{1}{ \sqrt{2} + \sqrt{3} - \sqrt{5}} =$

$=\dfrac{ \sqrt{2} + \sqrt{3} + \sqrt{5}}{ (\sqrt{2} + \sqrt{3} - \sqrt{5})(\sqrt{2} + \sqrt{3} + \sqrt{5})}\\ = \dfrac{ \sqrt{2} + \sqrt{3} + \sqrt{5}}{ {( \sqrt{2} + \sqrt{3} )}^{2} - 5} = \dfrac{ \sqrt{2} + \sqrt{3} + \sqrt{5}}{2 + 2 \sqrt{6} + 3 - 5} \\ = \dfrac{ \sqrt{2} + \sqrt{3} + \sqrt{5}}{2 \sqrt{6}} = \dfrac{ 2\sqrt{3} + 3\sqrt{2} + \sqrt{30}}{12}$