Question

Oii amores❤️faça

a fatoração dos números naturais a seguir verifique quais deles são números quadrados perfeitos?.

A) 2025 B) 252 C) 1296 D) 484 E) 441
F) 1024
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Answer 1

$\boxed{\begin{array}{l}\tt a)~\begin{array}{c|l}\sf2025&\sf3\\\sf675&\sf3\\\sf225&\sf3\\\sf75&\sf3\\\sf25&\sf5\\\sf5&\sf5\\\sf1\end{array}\\\sf 2025=3^4\cdot 5^2\implies quadrado~perfeito\\\tt b)~\begin{array}{c|l}\sf252&\sf2\\\sf126&\sf2\\\sf63&\sf3\\\sf 21&\sf3\\\sf7&\sf7\\\sf1\end{array}\\\sf 252=2^2\cdot3^2\cdot7\implies n\tilde ao ~\acute e~quadrado~perfeito\end{array}}$

$\boxed{\begin{array}{l}\tt c)~\begin{array}{c|l}\sf1296&\sf2\\\sf648&\sf2\\\sf324&\sf2\\\sf 162&\sf2\\\sf 81&\sf3\\\sf27&\sf3\\\sf9&\sf3\\\sf3&\sf3\\\sf1\end{array}\\\sf 1296=2^4\cdot3^4\implies \acute e~quadrado~perfeito\\\tt d)~\begin{array}{c|l}\sf484&\sf2\\\sf 242&\sf2\\\sf 121&\sf11\\\sf11&\sf11\\\sf1\end{array}\\\sf 484=2^2\cdot11^2\implies \acute e~quadrado~perfeito\end{array}}$

$\boxed{\begin{array}{l}\tt e)\begin{array}{c|l}\sf441&\sf3\\\sf147&\sf3\\\sf49&\sf7\\\sf7&\sf7\\\sf1\end{array}\\\sf 441=3^2\cdot7^2\implies \acute e~quadrado~perfeito\\\tt f)~\begin{array}{c|l}\sf1024&\sf2\\\sf512&\sf2\\\sf256&\sf2\\\sf128&\sf2\\\sf64&\sf2\\\sf32&\sf2\\\sf16&\sf2\\\sf8&\sf2\\\sf4&\sf2\\\sf2&\sf2\\\sf1\end{array}\\\sf 1024=2^{10}\implies \acute e~quadrado~perfeito\end{array}}$