Question

Usando a decomposição em fatores primos, calcule a raiz quadrada de:
64
a)
1225

b) 19.36​

Answer 1

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$\tt a)~\sf\sqrt{64}\\\begin{array}{c|l}\sf64&\sf2\\\sf32&\sf2\\\sf16&\sf2\\\sf8&\sf2\\\sf4&\sf2\\\sf2&\sf2\\\sf1\end{array}\\\sf 64=2^6\\\sf\sqrt{64}=\sqrt{2^6}=\diagup\!\!\!\!\!\sqrt{(2^3)^{\diagdown\!\!\!\!\!2}}\\\sf \sqrt{64}=2^3=8\checkmark$

$\tt b)~\sf\sqrt{1225}\\\begin{array}{c|l}\sf1225&\sf5\\\sf245&\sf5\\\sf49&\sf7\\\sf7&\sf7\\\sf1\end{array}\\\sf\sqrt{1225}=\sqrt{5^{\diagup\!\!\!\!2}\cdot7^{\diagup\!\!\!\!2}}=5\cdot7=35\checkmark$

$\tt c)~\sf\sqrt{1936}\\\begin{array}{c|l}\sf1936&\sf2\\\sf968&\sf2\\\sf484&\sf2\\\sf242&\sf2\\\sf121&\sf11\\\sf11&\sf11\\\sf1\end{array}\\\sf\sqrt{1936}=\sqrt{2^{\diagup\!\!\!\!2}\cdot2^{\diagup\!\!\!\!\!2}\cdot11^{\diagup\!\!\!\!\!2}}\\\sf\sqrt{1936}=2\cdot2\cdot11\\\huge\boxed{\boxed{\boxed{\boxed{\sf\sqrt{1936}=44\checkmark }}}}$