Question

(Colégio Naval - Rj ) O valor de raíz terça de 16 raíz de 8 vezes raíz sexta de 0,125 é :

Answer 1

$\ \ \sqrt[3]{16\cdot \sqrt{8} } \cdot \sqrt[6]{0,125} \\ \\=\sqrt[3]{2^4\cdot \sqrt{2^3} } \cdot \sqrt[6]{ \frac{125^{:125}}{1000_{:125}} } \\ \\=\sqrt[3]{2^4\cdot 2^\frac{3}{2}} \cdot \sqrt[6]{ \frac{1}{8} } \\ \\=\sqrt[3]{2^{4+\frac{3}{2}}} \cdot \sqrt[6]{\big(\frac{1}{2}\big)^3} \\ \\=\sqrt[3]{2^{4+\frac{3}{2}}} \cdot \sqrt[6]{ 2^{-3}} \\ \\=\sqrt[3]{2^{\frac{8}{2}+\frac{3}{2}}} \cdot 2^{-\frac{3}{6}}\\ \\=\sqrt[3]{2^{\frac{8+3}{2}}} \cdot 2^{-\frac{3}{6}}$
$\\=\sqrt[3]{2^{\frac{11}{2}}} \cdot 2^{-\frac{3}{6}}\\ \\=\big(2^{\frac{11}{2}}\big)^\frac{1}{3} \cdot 2^{-\frac{3}{6}}\\ \\=2^{\frac{11}{2}\cdot\frac{1}{3}} \cdot 2^{-\frac{3}{6}}\\ \\=2^\frac{11}{6} \cdot 2^{-\frac{3}{6}}\\ \\=2^{\frac{11}{6} +\big(-\frac{3}{6}\big)}\\ \\=2^{\frac{11}{6} -\frac{3}{6}}\\ \\=2^{\frac{11-3}{6}}\\ \\=2^\frac{8^{:2}}{6_{:2}}\\ \\=2^\frac{4}{3}$
$=2^\frac{4}{3}\\ \\= \sqrt[3]{2^4} \\ \\= \sqrt[3]{2^3\cdot2^1} \\ \\= \sqrt[3]{2^3} \cdot \sqrt[3]{2}\\ \\=2\sqrt[3]{2}$