1-t-u-v-t-u-v-t-u-v-t-u-v-t-u-v-t-u-v-t-u-v-t-u-v-t-3-u-v-t-2-u-v-2-t-u-v-u-v-t-4-2-u-v-t-2-u-v-

Question Number 70780 by MJS last updated on 08/Oct/19

\frac{1}{t + \sqrt{u} + \sqrt{v}} =

= \frac{(t - \sqrt{u} + \sqrt{v}) (t + \sqrt{u} - \sqrt{v}) (t - \sqrt{u} - \sqrt{v})}{(t + \sqrt{u} + \sqrt{v}) (t - \sqrt{u} + \sqrt{v}) (t + \sqrt{u} - \sqrt{v}) (t - \sqrt{u} - \sqrt{v})} =

= \frac{t^{3} - (\sqrt{u} + \sqrt{v}) t^{2} - {(\sqrt{u} - \sqrt{v})}^{2} t + (u - v) (\sqrt{u} - \sqrt{v})}{t^{4} - 2 (u + v) t^{2} + {(u - v)}^{2}}

\frac{1}{t + \sqrt[3]{u} + \sqrt[3]{v}} =

| \begin{matrix} α = - \frac{1}{2} + \frac{\sqrt{3}}{2} i; β = - \frac{1}{2} - \frac{\sqrt{3}}{2} i \Rightarrow \\ \Rightarrow α^{2} = β; β^{2} = α; α + β = - 1; α β = 1 \end{matrix} |

= \frac{(t + α \sqrt[3]{u} + β \sqrt[3]{v}) (t + β \sqrt[3]{u} + α \sqrt[3]{v})}{(t + \sqrt[3]{u} + \sqrt[3]{v}) (t + α \sqrt[3]{u} + β \sqrt[3]{v}) (t + β \sqrt[3]{u} + α \sqrt[3]{v})} =

= \frac{t^{2} - (\sqrt[3]{u} + \sqrt[3]{v}) t + \sqrt[3]{u^{2}} - \sqrt[3]{u v} + \sqrt[3]{v^{2}}}{t^{3} + u + v - 3 \sqrt[3]{u v} t} =

| \begin{matrix} a = t^{3} + u + v; b = 27 {u v t}^{3} \\ \frac{1}{a - \sqrt[3]{b}} = \frac{(α a - β \sqrt[3]{b}) (β a - α \sqrt[3]{b})}{(a - \sqrt[3]{b}) (α a - β \sqrt[3]{b}) (β a - α \sqrt[3]{b})} = \\ = \frac{a^{2} + a \sqrt[3]{b} + \sqrt[3]{b^{2}}}{a^{3} - b} \end{matrix} |

= \frac{N}{t^{9} + 3 (u + v) t^{6} + 3 (u^{2} - 7 u v + v^{2}) t^{3} + {(u + v)}^{3}}

N =

= t^{8} -

- (\sqrt[3]{u} + \sqrt[3]{v}) t^{7} +

+ {(\sqrt[3]{u} + \sqrt[3]{v})}^{2} t^{6} +

+ (\sqrt[3]{u} + \sqrt[3]{v}) (\sqrt[3]{u} - 2 \sqrt[3]{v}) (2 \sqrt[3]{u} - \sqrt[3]{v}) t^{5} -

- {(\sqrt[3]{u} + \sqrt[3]{v})}^{2} (\sqrt[3]{u} - 2 \sqrt[3]{v}) (2 \sqrt[3]{u} - \sqrt[3]{v}) t^{4} +

+ {(\sqrt[3]{u} + \sqrt[3]{v})}^{3} (\sqrt[3]{u} - 2 \sqrt[3]{v}) (2 \sqrt[3]{u} - \sqrt[3]{v}) t^{3} +

+ {(\sqrt[3]{u^{2}} - \sqrt[3]{u v} + \sqrt[3]{v^{2}})}^{3} t^{2} -

- (\sqrt[3]{u} + \sqrt[3]{v}) {(\sqrt[3]{u^{2}} - \sqrt[3]{u v} + \sqrt[3]{v^{2}})}^{3} t +

+ {(\sqrt[3]{u} + \sqrt[3]{v})}^{2} {(\sqrt[3]{u^{2}} - \sqrt[3]{u v} + \sqrt[3]{v^{2}})}^{3}

I think {there}^{'} s no easier way \dots

Commented by MJS last updated on 08/Oct/19

this had been requested (but deleted)

\frac{1}{1 + 2 \sqrt[3]{3} + 3 \sqrt[3]{2}} =

= \frac{- 8208 \sqrt[3]{36} + 2988 \sqrt[3]{18} - 378 \sqrt[3]{12} + 23020 \sqrt[3]{9} - 36024 \sqrt[3]{6} + 54225 \sqrt[3]{4} + 13114 \sqrt[3]{3} - 1659 \sqrt[3]{2} - 5423}{458047}

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