Question

Transforme o radical duplo abaixo em uma soma de radicais simples.

$\sqrt{ \frac{30(3 + \sqrt{5)} }{11} }$
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Answer 1

Resposta:

$\textsf{Leia abaixo}$

Explicação passo a passo:

$\mathsf{\sqrt{A \pm \sqrt{B}} = \sqrt{\dfrac{A + C}{2}} \pm \sqrt{\dfrac{A - C}{2}}}$

$\mathsf{\sqrt{\dfrac{30(3 + \sqrt{5})}{11}}}$

$\mathsf{\sqrt{\dfrac{90 + 30\sqrt{5}}{11}}}$

$\mathsf{\sqrt{\dfrac{90}{11} + \dfrac{30\sqrt{5}}{11}}}$

$\mathsf{\sqrt{\dfrac{90}{11} + \sqrt{\dfrac{30^2.5}{11^2}}}}$

$\mathsf{\sqrt{\dfrac{90}{11} + \sqrt{\dfrac{900.5}{121}}}}$

$\mathsf{\sqrt{\dfrac{90}{11} + \sqrt{\dfrac{4.500}{121}}}}$

$\mathsf{A = \dfrac{90}{11}}$

$\mathsf{B = \dfrac{4.500}{121}}$

$\mathsf{C = \sqrt{A^2 - B}}$

$\mathsf{C = \sqrt{\dfrac{8.100}{121} - \dfrac{4.500}{121}}}$

$\mathsf{C = \sqrt{\dfrac{3.600}{121}}}$

$\mathsf{C = \dfrac{60}{11}}$

$\mathsf{\sqrt{\dfrac{30(3 + \sqrt{5})}{11}} = \sqrt{\dfrac{90/11 + 60/11}{2}} + \sqrt{\dfrac{90/11 - 60/11}{2}}}$

$\mathsf{\sqrt{\dfrac{30(3 + \sqrt{5})}{11}} = \sqrt{\dfrac{150/11}{2}} + \sqrt{\dfrac{30/11}{2}}}$

$\mathsf{\sqrt{\dfrac{30(3 + \sqrt{5})}{11}} = \sqrt{\dfrac{150}{22}} + \sqrt{\dfrac{30}{22}}}$

$\boxed{\boxed{\mathsf{\sqrt{\dfrac{30(3 + \sqrt{5})}{11}} = \sqrt{\dfrac{75}{11}} + \sqrt{\dfrac{15}{11}}}}}$