Question

Simplifique a expressão:

n-1√a/n√a

Answer 1

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Simplificar a expressão:

$\mathsf{\,^{n-1}\hspace{-6}\sqrt{\dfrac{a}{\,^n\hspace{-5}\sqrt{a}}}}\qquad\quad\textsf{(para }\mathsf{a\ \textgreater \ 0~~e~~n>1,\,n\in\mathbb{N}}\textsf{)}\\\\\\ =\mathsf{\,^{n-1}\hspace{-6}\sqrt{\dfrac{\,^n\hspace{-5}\sqrt{a^n}}{\,^n\hspace{-5}\sqrt{a}}}}\qquad\quad\textsf{(pois }\mathsf{a=\,^n\hspace{-5}\sqrt{a^n}}\textsf{)}\\\\\\ =\mathsf{\,^{n-1}\hspace{-6}\sqrt{\,^n\hspace{-7}\sqrt{\dfrac{a^n}{a}}}}\\\\\\ =\mathsf{\,^{n-1}\hspace{-6}\sqrt{\,^n\hspace{-5}\sqrt{a^{n-1}}}}$

$=\mathsf{\,^{n-1}\hspace{-6}\sqrt{a^{\hspace{-5}\footnotesize\begin{array}{l}\mathsf{\frac{n-1}{n}} \end{array}}}}\\\\ =\mathsf{\left(a^{\hspace{-5}\footnotesize\begin{array}{l}\mathsf{\frac{n-1}{n}} \end{array}\hspace{-5}}\right)^{\hspace{-5}\footnotesize\begin{array}{l}\mathsf{\frac{1}{n-1}}\end{array}}}$

$=\mathsf{a^{\hspace{-5}\footnotesize\begin{array}{l}\mathsf{\frac{n-1}{n}\cdot \frac{1}{n-1}} \end{array}\hspace{-5}}}\\\\ =\mathsf{a^{\hspace{-5}\footnotesize\begin{array}{l}\mathsf{\frac{1}{n}} \end{array}\hspace{-5}}}\\\\ =\mathsf{\,^n\hspace{-5}\sqrt{a}}\quad\longleftarrow\quad\textsf{esta \'e a resposta.}$


Bons estudos! :-)


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