Exercício sobre álgebra fundamental

Escrever o radical $\phantom{X}\sqrt{2\,\centerdot\,\sqrt[\large 3]{2}}\phantom{X}$ na forma de uma potência de expoente racional.

resposta:

Resolução:

MODO 1.
$\,\sqrt{2\,\centerdot\,\sqrt[\large 3]{2}}\,=\,\sqrt{2}\,\centerdot\,\sqrt{\sqrt[3]{2}}\,$ $=\,\sqrt{2}\,\centerdot\,\sqrt[\large 6]{2}\,$ $=\,2^{\large \frac{1}{2}}\,\centerdot \,2^{\large \frac{1}{6}}\,$ $=\,2^{\frac{1}{2}\,+\,\frac{1}{6}}\,$ $=\,2^{\frac{4}{6}}\,=\,2^{\frac{2}{3}}\,$

MODO 2.
$\,\sqrt{2\,\centerdot\,\sqrt[\large 3]{2}}\,=\,\sqrt{2}\,\centerdot\,\sqrt{\sqrt[3]{2}}\,$ $=\,\sqrt[\large 2]{2^1}\,\centerdot\,\sqrt[\large 6]{2^1}\,$ $=\,\sqrt[\large 6]{2^3}\,\centerdot\,\sqrt[\large 6]{2^1}\,$ $=\,\sqrt[\large 6]{2^3\,\centerdot\,2^1}\,$ $=\,\sqrt[\large 6]{2^4}\,=\,\sqrt[\large 3]{2^2}\,=\,2^{\frac{2}{3}}\,$

MODO 3.
Partir da observação seguinte: $\,2\,=\,\sqrt[\large 3]{2^3}\,$
$\,\sqrt{2\,\centerdot\,\sqrt[\large 3]{2}}\,=\,\sqrt{\sqrt[\large 3]{2^3}\,\centerdot\,\sqrt[\large 3]{2}}\,$ $=\,\sqrt[\large 2]{\sqrt[\large 3]{2^3\,\centerdot\,2}}\,\,$ $=\,\sqrt[\large 6]{2^4}\,$ $=\,\sqrt[\large 3]{2^2}\,=\,2^{\frac{2}{3}}\,$

Resposta:

$\;2^{2/3}\;$
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