Question

retire fatores do radicando (a) √48 (b) √720

Answer 1

(a) Decompondo $48$ em seus fatores primos:

$\begin{array}{r|l} 48&2\\ 24&2\\ 12&2\\ 6&2\\ 3&3\\ 1 \end{array}~~\Rightarrow~~48=2^{4}\cdot 3$


Portanto,

$\sqrt{48}\\ \\ =\sqrt{2^{4}\cdot 3}\\ \\ =\sqrt{2^{2\,\cdot\,2}\cdot 3}\\ \\ =\sqrt{(2^{2})^{2}\cdot 3}\\ \\ =\sqrt{(2^{2})^{2}}\cdot \sqrt{3}\\ \\ =2^{2}\cdot \sqrt{3}\\ \\ =4\sqrt{3}$


(b) Decompondo $720$ em seus fatores primos:

$\begin{array}{r|l} 720&2\\ 360&2\\ 180&2\\ 90&2\\ 45&3\\ 15&3\\ 5&5\\ 1 \end{array}~~\Rightarrow~~720=2^{4}\cdot 3^{2}\cdot 5$


Portanto,

$\sqrt{720}\\ \\ =\sqrt{2^{4}\cdot 3^{2}\cdot 5}\\ \\ =\sqrt{2^{2\,\cdot\,2}\cdot 3^{2}\cdot 5}\\ \\ =\sqrt{(2^{2})^{2}\cdot 3^{2}\cdot 5}\\ \\ =\sqrt{(2^{2}\cdot 3)^{2}\cdot 5}\\ \\ =\sqrt{(2^{2}\cdot 3)^{2}}\cdot \sqrt{5}\\ \\ =2^{2}\cdot 3\cdot \sqrt{5}\\ \\ =12\sqrt{5}$