Question

Simplificando a expressão $\mathsf{\sqrt{\dfrac{600}{25^{n+2}-5^{2n+2}}}}$ para n pertencente aos naturais :

a) 5
b) 5^-1
c) 5^-2
d) 5^2
e) 5^0

foto anexada

Answer 1

Caso esteja usando o app, experimente abrir esta resposta pelo navegador:  https://brainly.com.br/tarefa/8199730

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Simplificar a expressão:

$\mathsf{E=\,^n\hspace{-6}\sqrt{\dfrac{600}{25^{n+2}-5^{2n+2}}}\qquad\qquad n\in\mathbb{N}^*}\\\\\\ \mathsf{E=\,^n\hspace{-6}\sqrt{\dfrac{600}{(5^2)^{n+2}-5^{2n+2}}}}\\\\\\ \mathsf{E=\,^n\hspace{-6}\sqrt{\dfrac{600}{5^{2\cdot (n+2)}-5^{2n+2}}}}\\\\\\ \mathsf{E=\,^n\hspace{-6}\sqrt{\dfrac{600}{5^{2n+4}-5^{2n+2}}}}$

$\mathsf{E=\,^n\hspace{-6}\sqrt{\dfrac{600}{5^{2n+2+2}-5^{2n+2}}}}\\\\\\ \mathsf{E=\,^n\hspace{-6}\sqrt{\dfrac{600}{5^{2n+2}\cdot 5^2-5^{2n+2}}}}\\\\\\ \mathsf{E=\,^n\hspace{-6}\sqrt{\dfrac{600}{5^{2n+2}\cdot 25-5^{2n+2}}}}$


Coloque $\mathsf{5^{2n+2}}$ em evidência no denominador, e a expressão fica

$\mathsf{E=\,^n\hspace{-6}\sqrt{\dfrac{600}{5^{2n+2}\cdot (25-1)}}}\\\\\\ \mathsf{E=\,^n\hspace{-6}\sqrt{\dfrac{600}{5^{2n+2}\cdot 24}}}\\\\\\ \mathsf{E=\,^n\hspace{-6}\sqrt{\dfrac{25\cdot \diagup\hspace{-9}24}{5^{2n+2}\cdot \diagup\hspace{-9}24}}}\\\\\\ \mathsf{E=\,^n\hspace{-6}\sqrt{\dfrac{5^2}{5^{2n+2}}}}$

$\mathsf{E=\,^{^{\footnotesize\begin{array}{l} \mathsf{n} \end{array}}}\hspace{-11}\sqrt{5^{2-(2n+2)}}}\\\\ \mathsf{E=\,^{^{\footnotesize\begin{array}{l} \mathsf{n} \end{array}}}\hspace{-11}\sqrt{5^{2-2n-2}}}\\\\ \mathsf{E=\,^{^{\footnotesize\begin{array}{l} \mathsf{n} \end{array}}}\hspace{-11}\sqrt{5^{-2n}}}\\\\ \mathsf{E=(5^{-2n})^{1/n}}\\\\ \mathsf{E=5^{-2}}\quad\longleftarrow\quad\textsf{esta \'e a resposta (alternativa c).}$


Bons estudos! :-)


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