Show-that-5-2-3-1-4-is-in-F-2-by-expressing-the-number-in-form-a-1-b-1-k-1-where-a-1-b-1-k-1-are-in-F-1-

Question Number 133072 by bemath last updated on 18/Feb/21

Show that \frac{5}{2 - \sqrt[4]{3}} is in F_{2} by expressing

the number in form a_{1} + b_{1} \sqrt{k_{1}} where

a_{1}, b_{1}, k_{1} a r e i n F_{1}

Answered by EDWIN88 last updated on 18/Feb/21

\frac{5}{2 - \sqrt[4]{3}} \times \frac{2 + \sqrt[4]{3}}{2 + \sqrt[4]{3}} = \frac{10 + 5 \sqrt[4]{3}}{4 - \sqrt[4]{9}} \times \frac{4 + \sqrt[4]{9}}{4 + \sqrt[4]{9}}

= \frac{40 + 10 \sqrt[4]{9} + 20 \sqrt[4]{3} + 5 \sqrt[4]{27}}{13}

= \frac{40 + 10 \sqrt{3} + 20 \sqrt[4]{3} + 5 \sqrt{3} \sqrt[4]{3}}{13}

= \frac{40 + 10 \sqrt{3}}{13} + (\frac{20 + 5 \sqrt{3}}{13}) \sqrt[4]{3}

then a_{1} = \frac{40 + 10 \sqrt{3}}{13}; b_{1} = \frac{20 + 5 \sqrt{3}}{13}; k_{1} = \sqrt{3}

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