Operaciones con radicales [2854]

, por dani

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– a) \sqrt[3]{512} : \sqrt[3]{200}
\left. \begin{array}{c|c}512 & 2\cr256 & 2 \cr128 & 2 \cr64 & 2 \cr32 & 2 \cr16 & 2 \cr8 & 2 \cr4 & 2 \cr2 & 2 \cr1\end{array} \right. \qquad \qquad \left. \begin{array}{c|c}200 & 2\cr100 & 2 \cr50 & 2 \cr25 & 5 \cr5 & 5 \cr1\end{array} \right.
\sqrt[3]{512} : \sqrt[3]{200} = \frac{\sqrt[3]{512}}{\sqrt[3]{200}} = \sqrt[3]{\frac{512}{200}} = \sqrt[3]{\frac{2^9}{2^3 \cdot 5^2}} =
\sqrt[3]{\frac{\cancel{2^3} \cdot 2^3 \cdot 2^3}{\cancel{2^3} \cdot 5^2}} =
2 \cdot 2 \cdot \sqrt[3]{\frac{1}{ 5^2}} = 4 \sqrt[3]{\frac{1}{25}}

– b) \sqrt[4]{2187} : \sqrt{108} = \frac{\sqrt[4]{2187}}{\sqrt{108}}
Para dividir radicales de distinto índice debemos hacer índice común
\frac{\sqrt[4]{2187}}{\sqrt{108}} = \frac{\sqrt[4]{2187}}{\sqrt[4]{(108)^2}} = \sqrt[4]{\frac{2187}{(108)^2}} = \sqrt[4]{\frac{3^7}{(2^2 \cdot 3^3)^2}} =
\sqrt[4]{\frac{3^7}{2^4 \cdot 3^6}} =\sqrt[4]{\frac{3}{2^4}} = \frac{\sqrt[4]{3}}{\sqrt[4]{2^4}} = \frac{\sqrt[4]{3}}{2}
\left. \begin{array}{c|c}2187 & 3\cr729 & 3 \cr243 & 3 \cr81 & 3 \cr27 & 3 \cr9 & 3 \cr3 & 3 \cr1\end{array} \right. \qquad \qquad \left. \begin{array}{c|c}108 & 2\cr54 & 2 \cr27 & 3 \cr9 & 3 \cr3 & 3 \cr1\end{array} \right.

Realiza las siguientes operaciones:

– a) \sqrt[3]{512} : \sqrt[3]{200}
– b) \sqrt[4]{2187} : \sqrt{108}