Racionalize corretamente o denominador de![\frac{1+ \sqrt[3]{2} }{1+ \sqrt[3]{2} + \sqrt[3]{4} }](https://tex.z-dn.net/?f=+%5Cfrac%7B1%2B+%5Csqrt%5B3%5D%7B2%7D+%7D%7B1%2B+%5Csqrt%5B3%5D%7B2%7D+%2B+%5Csqrt%5B3%5D%7B4%7D+%7D+ "\frac{1+ \sqrt[3]{2} }{1+ \sqrt[3]{2} + \sqrt[3]{4} }")
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1 + ∛2 + ∛4 =
1 +∛2 + ∛2²
1 + ∛2 + ∛4 = - 1/(1 - ∛2 )
( 1 + ∛2 ) / ( 1 + ∛2 + ∛4)
( 1 + ∛2 ) /1 * ( 1 - ∛2)/-1
1 - ( ∛2)² / ( -1)
(1 - ∛4) / -1
∛4 - 1 ****
DanJR:
Não consegui entender o que foi feito no denominador da segunda linha!
Faz parte. Acontece com qualquer um! Até!!
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Olá!
Sabemos que
.
Portanto,
![\\(1-2)=(1^{\frac{1}{3}}-2^{\frac{1}{3}})(1^{\frac{2}{3}}+1^{\frac{1}{3}}\cdot2^{\frac{1}{3}}+2^{\frac{2}{3}})\\\\ - 1 = (1 - \sqrt[3]{2})(1 + \sqrt[3]{2} + \sqrt[3]{2^2}) \\\\ 1 + \sqrt[3]{2} + \sqrt[3]{4} = - \frac{1}{1 - \sqrt[3]{2}}](https://tex.z-dn.net/?f=%5C%5C%281-2%29%3D%281%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D-2%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D%29%281%5E%7B%5Cfrac%7B2%7D%7B3%7D%7D%2B1%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D%5Ccdot2%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D%2B2%5E%7B%5Cfrac%7B2%7D%7B3%7D%7D%29%5C%5C%5C%5C+-+1+%3D+%281+-+%5Csqrt%5B3%5D%7B2%7D%29%281+%2B+%5Csqrt%5B3%5D%7B2%7D+%2B+%5Csqrt%5B3%5D%7B2%5E2%7D%29+%5C%5C%5C%5C+1+%2B+%5Csqrt%5B3%5D%7B2%7D+%2B+%5Csqrt%5B3%5D%7B4%7D+%3D+-+%5Cfrac%7B1%7D%7B1+-+%5Csqrt%5B3%5D%7B2%7D%7D "\\(1-2)=(1^{\frac{1}{3}}-2^{\frac{1}{3}})(1^{\frac{2}{3}}+1^{\frac{1}{3}}\cdot2^{\frac{1}{3}}+2^{\frac{2}{3}})\\\\ - 1 = (1 - \sqrt[3]{2})(1 + \sqrt[3]{2} + \sqrt[3]{2^2}) \\\\ 1 + \sqrt[3]{2} + \sqrt[3]{4} = - \frac{1}{1 - \sqrt[3]{2}}")
Substituindo,
![\\ \frac{1 + \sqrt[3]{2}}{1 + \sqrt[3]{2} + \sqrt[3]{4}} = \\\\\\ (1 + \sqrt[3]{2}) \div \frac{- 1}{1 - \sqrt[3]{2}} = \\\\\\ (1 + \sqrt[3]{2}) \times \frac{1 - \sqrt[3]{2}}{- 1} = \\\\\\ \frac{1 - (\sqrt[3]{2})^2}{- 1} = \\\\\\ \frac{1 - \sqrt[3]{4}}{- 1} = \\\\\\ \boxed{\boxed{\sqrt[3]{4} - 1}}](https://tex.z-dn.net/?f=%5C%5C+%5Cfrac%7B1+%2B+%5Csqrt%5B3%5D%7B2%7D%7D%7B1+%2B+%5Csqrt%5B3%5D%7B2%7D+%2B+%5Csqrt%5B3%5D%7B4%7D%7D+%3D+%5C%5C%5C%5C%5C%5C+%281+%2B+%5Csqrt%5B3%5D%7B2%7D%29+%5Cdiv+%5Cfrac%7B-+1%7D%7B1+-+%5Csqrt%5B3%5D%7B2%7D%7D+%3D+%5C%5C%5C%5C%5C%5C+%281+%2B+%5Csqrt%5B3%5D%7B2%7D%29+%5Ctimes+%5Cfrac%7B1+-+%5Csqrt%5B3%5D%7B2%7D%7D%7B-+1%7D+%3D+%5C%5C%5C%5C%5C%5C+%5Cfrac%7B1+-+%28%5Csqrt%5B3%5D%7B2%7D%29%5E2%7D%7B-+1%7D+%3D+%5C%5C%5C%5C%5C%5C+%5Cfrac%7B1+-+%5Csqrt%5B3%5D%7B4%7D%7D%7B-+1%7D+%3D+%5C%5C%5C%5C%5C%5C+%5Cboxed%7B%5Cboxed%7B%5Csqrt%5B3%5D%7B4%7D+-+1%7D%7D "\\ \frac{1 + \sqrt[3]{2}}{1 + \sqrt[3]{2} + \sqrt[3]{4}} = \\\\\\ (1 + \sqrt[3]{2}) \div \frac{- 1}{1 - \sqrt[3]{2}} = \\\\\\ (1 + \sqrt[3]{2}) \times \frac{1 - \sqrt[3]{2}}{- 1} = \\\\\\ \frac{1 - (\sqrt[3]{2})^2}{- 1} = \\\\\\ \frac{1 - \sqrt[3]{4}}{- 1} = \\\\\\ \boxed{\boxed{\sqrt[3]{4} - 1}}")
Espero ter ajudado!
Sabemos que
Portanto,
Substituindo,
Espero ter ajudado!
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