Simplify-5-2-1-3-5-2-1-3-

Question Number 45546 by Tawa1 last updated on 14/Oct/18

Simplify : \sqrt[3]{\sqrt{5} + 2} + \sqrt[3]{\sqrt{5} - 2}

Commented by Meritguide1234 last updated on 14/Oct/18

x^{3} = (\sqrt{5} + 2) + (\sqrt{5} - 2) + 3 . x

Answered by Meritguide1234 last updated on 14/Oct/18

x^{3} - 3 x - 2 \sqrt{5} = 0

Commented by Meritguide1234 last updated on 14/Oct/18

x (x^{2} - 3) = \sqrt{5} ({(\sqrt{5})}^{2} - 3)

x = \sqrt{5} (u n i q u e s o l u t i o n)

Answered by MJS last updated on 14/Oct/18

x = \sqrt[3]{2 + \sqrt{5}} - \sqrt[3]{2 - \sqrt{5}}

x^{3} = (2 + \sqrt{5}) - 3 {({(2 + \sqrt{5})}^{2} (2 - \sqrt{5}))}^{\frac{1}{3}} + 3 {((2 + \sqrt{5}) {(2 - \sqrt{5})}^{2})}^{\frac{1}{3}} - (2 - \sqrt{5})

x^{3} = 2 \sqrt{5} - 3 {((2 + \sqrt{5}) (- 1))}^{\frac{1}{3}} + 3 {((2 - \sqrt{5}) (- 1))}^{\frac{1}{3}}

x^{3} = 2 \sqrt{5} + 3 (\sqrt[3]{2 + \sqrt{5}} - \sqrt[3]{2 - \sqrt{5}})

x^{3} = 2 \sqrt{5} + 3 x

x^{3} - 3 x = 2 \sqrt{5}

in this special case {it}^{'} s easy to see that x_{1} = \sqrt{5}

because {(\sqrt{5})}^{3} = 5 \sqrt{5}

x^{3} - 3 x - 2 \sqrt{5} = (x - \sqrt{5}) (x^{2} + \sqrt{5} x + 2)

x_{2}, x_{3} \notin R

\Rightarrow x = \sqrt{5}

btw . : {(\frac{1}{2} + \frac{\sqrt{5}}{2})}^{3} = 2 + \sqrt{5} {(- \frac{1}{2} + \frac{\sqrt{5}}{2})}^{3} = - 2 + \sqrt{5}

Commented by Tawa1 last updated on 14/Oct/18

God bless you sir

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