Question

Alguém

utilizando os conhecimentos adquiridos Sobre a fatoração e radiciação encontre as seguintes raízes:

$a)\sqrt[3]{27}$
$b) \sqrt[3]{343}$
$c) \sqrt[9]{512}$
$d) \sqrt[10]{1024}$
$e) \: \sqrt[2]{576}$

utilizando os conhecimentos adquiridos sobre fatoração e radiciação encontre as seguintes raízes:

$a) \sqrt[5]{486} =$
$b) \sqrt[3]{540 } =$
$c) \: \sqrt{600} =$
$d) \: \sqrt[8]{256} =$

​

Answer 1

$\boxed{\begin{array}{l}\rm 1)\\\tt a )~\begin{array}{c|l}\sf27&\sf3\\\sf9&\sf3\\\sf3&\sf3\\\sf1\end{array}\\\sf 27=3^3\\\sf\sqrt[\sf3]{\sf27}=\sqrt[\sf3]{\sf 3^3}=3\\\tt b)~\begin{array}{c|l}\sf343&\sf3\\\sf81&\sf3\\\sf27&\sf3\\\sf9&\sf3\\\sf3&\sf3\\\sf1\end{array}\\\sf 343=3^3\cdot3^2\\\sf \sqrt[\sf3]{\sf 3^3\cdot3^2}=3\sqrt[\sf3]{\sf 3^2}=3\sqrt[\sf3]{\sf9}\end{array}}$

$\boxed{\begin{array}{l}\tt c)\\\begin{array}{c|l}\sf512&\sf2\\\sf256&\sf2\\\sf128&\sf2\\\sf64&\sf2\\\sf32&\sf2\\\sf16&\sf2\\\sf8&\sf2\\\sf4&\sf2\\\sf2&\sf2\\\sf1\end{array}\\\sf \sqrt[\sf9]{\sf 512}=\sqrt[\sf9]{\sf2^9}=2\end{array}}$

$\boxed{\begin{array}{l}\tt d)\\\begin{array}{c|l}\sf1024&\sf2\\\sf512&\sf2\\\sf256&\sf2\\\sf128&\sf2\\\sf64&\sf2\\\sf32&\sf2\\\sf16&\sf2\\\sf8&\sf2\\\sf4&\sf2\\\sf2&\sf2\\\sf1\end{array}\\\sf \sqrt[\sf10]{\sf 1024}=\sqrt[\sf10]{\sf2^{10}}=2\end{array}}$

$\boxed{\begin{array}{l}\tt e)\\\begin{array}{c|l}\sf576&\sf2\\\sf288&\sf2\\\sf144&\sf2\\\sf72&\sf2\\\sf36&\sf2\\\sf18&\sf2\\\sf9&\sf3\\\sf3&\sf3\\\sf1\end{array}\\\sf 576=2^6\cdot3^2\\\sf\sqrt{576}=\sqrt{2^6\cdot3^2}=2^3\cdot3=8\cdot3=24\end{array}}$

$\boxed{\begin{array}{l}\rm 2)\\\tt a)\\\begin{array}{c|l}\sf486&\sf2\\\sf243&\sf3\\\sf81&\sf3\\\sf27&\sf3\\\sf9&\sf3\\\sf3&\sf3\\\sf1\end{array}\\\sf 486=2\cdot3^5\\\sf\sqrt[\sf5]{\sf 486}=\sqrt[\sf5]{\sf 2\cdot3^5}=3\sqrt[\sf5]{\sf2}\end{array}}$

$\boxed{\begin{array}{l}\tt b)\\\begin{array}{c|l}\sf540&\sf2\\\sf270&\sf2\\\sf135&\sf3\\\sf45&\sf3\\\sf15&\sf3\\\sf5&\sf5\\\sf1\end{array}\\\sf 540=2^2\cdot3^3\cdot5\\\sf\sqrt[\sf3]{\sf540}=\sqrt[\sf3]{\sf2^2\cdot3^3\cdot5}=3\sqrt[\sf3]{\sf4\cdot5}=3\sqrt[\sf3]{\sf20}\end{array}}$

$\boxed{\begin{array}{l}\tt d)\\\begin{array}{c|l}\sf600&\sf2\\\sf300&\sf2\\\sf150&\sf2\\\sf75&\sf3\\\sf25&\sf5\\\sf5&\sf5\\\sf1\end{array}\\\sf 600=2^2\cdot2\cdot3\cdot5^2\\\sf\sqrt{600}=\sqrt{2^2\cdot2\cdot3\cdot5^2}=2\cdot5\sqrt{2\cdot3}=10\sqrt{6}\end{array}}$

$\boxed{\begin{array}{l}\tt d)\\\begin{array}{c|l}\sf256&\sf2\\\sf128&\sf2\\\sf64&\sf2\\\sf32&\sf2\\\sf16&\sf2\\\sf8&\sf2\\\sf4&\sf2\\\sf2&\sf2\\\sf1\end{array}\\\sf256=2^8\\\sf\sqrt[\sf8]{\sf256}=\sqrt[\sf8]{\sf2^8}=2\end{array}}$

$\boxed{\begin{array}{l}\displaystyle\sf \ell ife=\int_{birthday}^{death}\dfrac{happiness}{time}d_{time}\end{array}}$