Question

Resolver las siguientes operaciones combinadas:

a) $\frac{\sqrt{5}-1 }{(1+\sqrt{5})^{2} }=$

b) $\frac{(3-\sqrt{3})^{2} }{(1+\sqrt{3})^{2} }=$

c) $\frac{\sqrt{3}+\sqrt{5} }{\sqrt{5}-\sqrt{3} }-2\sqrt{5}.(4\sqrt{5}-3\sqrt{3})=$

Answer 1

Respuesta:

Explicación paso a paso:

Antes de empezar , hay que recordar esto:

(a+b)^2 = a^2 + 2ab + b^2

(a-b)^2 = a^2 -2ab + b^2

(a+b ) * ( a-b)  = a ^2 - b ^2

a) $\frac{\sqrt{5} -1}{(1+\sqrt{5})^{2} } = \frac{\sqrt{5}- 1}{1 +( \sqrt{5})^{2} + 2\sqrt{5} }= \frac{\sqrt{5} -1}{1 +5 + 2 \sqrt{5} } = \frac{\sqrt{5} -1}{6+2\sqrt{5} } =\frac{\sqrt{5}-1 }{6 + 2 \sqrt{5} } *\frac{6-2\sqrt{5} }{6-2\sqrt{5} } = \frac{6\sqrt{5} -2\sqrt{5} *\sqrt{5} -6 + 2\sqrt{5} }{6^{2} - (2\sqrt{5})^{2} } = \frac{6\sqrt{5}-2*5-6+2\sqrt{5} }{36 - 4 * 5} = \frac{-16 +8\sqrt{5} }{16} = -1 + \frac{\sqrt{5} }{2}$

b) $\frac{(3-\sqrt{3})^{2} }{(1+\sqrt{3})^{2} } = \frac{3^{2} +(\sqrt{3})^{2} - 2* 3*\sqrt{3} }{1 + (\sqrt{3})^{2}+ 2*1*\sqrt{3} } = \frac{9 + 3 -6\sqrt{3} }{1+3+2\sqrt{3} } = \frac{12 -6\sqrt{3} }{4+2\sqrt{3} } = \frac{12 -6\sqrt{3} }{4+2\sqrt{3} } * \frac{4-2\sqrt{3} }{4-2\sqrt{3}} = \frac{48 - 24\sqrt{3} -24\sqrt{3}-6*-2*\sqrt{3} *\sqrt{3} }{4^{2} - (2\sqrt{3} )^{2} }\\\frac{x}{y} = \frac{48 + 36 -24\sqrt{3} -24\sqrt{3} }{16-12} =\\= \frac{84 - 48\sqrt{3} }{4} = 21 - 12\sqrt{3}$