Question

calcule o valor de:
$\sqrt{5000 + \sqrt{500} }$
​

Answer 1

$\ \ \ \sqrt{5000} +\sqrt{500} \\\\=\sqrt{100\cdot25\cdot 2} +\sqrt{100\cdot5} \\\\=\sqrt{100}\cdot\sqrt{25}\cdot \sqrt{2} +\sqrt{100}\cdot\sqrt{5} \\\\=10\cdot5\cdot \sqrt{2} +10\cdot\sqrt{5} \\\\=10\cdot\Big(5 \sqrt{2} +\sqrt{5}\Big)$

- - - - - - - - - - - - - - - - - - - - - - - - -

$5000=2^3\cdot5^4 \ \ \ e \ \ \ 500=2^2\cdot5^3\\\\ \ \ \sqrt{5000} +\sqrt{500} \\\\=\sqrt{2^3\cdot5^4} +\sqrt{2^2\cdot5^3} \\\\=\sqrt{2^2 \cdot 2^1\cdot5^4} +\sqrt{2^2\cdot5^2\cdot5^1} \\\\=\sqrt{2^2 \cdot 5^4\cdot2^1} +\sqrt{2^2\cdot5^2\cdot5^1} \\\\=\sqrt{2^2} \cdot \sqrt{5^4}\cdot\sqrt{2^1} +\sqrt{2^2}\cdot\sqrt{5^2}\cdot\sqrt{5^1} \\\\=2\cdot 5^2\cdot\sqrt{2} +2\cdot5\cdot\sqrt{5} \\\\=2\cdot 25\cdot\sqrt{2} +10\cdot\sqrt{5} \\\\=50\cdot\sqrt{2} +10\cdot\sqrt{5}$

$\\=10\cdot\Big(5\sqrt{2} +\sqrt{5}\Big)$