Question

Resolva as seguintes expressões:

a- ∛625
b- ⁴√512
c-∛256
d-⁴√1024
e- ⁴√7⁵.7³
f) ¹⁰√2¹⁵ . 3⁸

Answer 1

a) $\,^{3}\!\!\!\sqrt{625}$

$=\,^{3}\!\!\!\sqrt{5^{4}}\\ \\ =\,^{3}\!\!\!\sqrt{5^{3+1}}\\ \\ =\,^{3}\!\!\!\sqrt{5^{3}\cdot 5^{1}}\\ \\ =\,^{3}\!\!\!\sqrt{5^{3}\cdot 5}\\ \\ =\,^{3}\!\!\!\sqrt{5^{3}}\cdot \,^{3}\!\!\!\sqrt{5}\\ \\ =5\,^{3}\!\!\!\sqrt{5}$


b) $\,^{4}\!\!\!\sqrt{512}$

$=\,^{4}\!\!\!\sqrt{2^{9}}\\ \\ =\,^{4}\!\!\!\sqrt{2^{8}\cdot 2^{1}}\\ \\ =\,^{4}\!\!\!\sqrt{2^{2\,\cdot\, 4}\cdot 2}\\ \\ =\,^{4}\!\!\!\!\sqrt{(2^{2})^{4}\cdot 2}\\ \\ =\,^{4}\!\!\!\!\sqrt{(2^{2})^{4}}\cdot \,^{4}\!\!\!\sqrt{2}\\ \\ =2^{2}\cdot \,^{4}\!\!\!\sqrt{2}\\ \\ =4\,^{4}\!\!\!\sqrt{2}$


c) $\,^{3}\!\!\!\sqrt{256}$

$=\,^{3}\!\!\!\sqrt{2^{8}}\\ \\ =\,^{3}\!\!\!\sqrt{2^{6+2}}\\ \\ =\,^{3}\!\!\!\sqrt{2^{6}\cdot 2^{2}}\\ \\ =\,^{3}\!\!\!\sqrt{2^{2\,\cdot\,3}\cdot 2^{2}}\\ \\ =\,^{3}\!\!\!\!\sqrt{(2^{2})^{3}\cdot 2^{2}}\\ \\ =\,^{3}\!\!\!\!\sqrt{(2^{2})^{3}\cdot 4}\\ \\ =\,^{3}\!\!\!\!\sqrt{(2^{2})^{3}}\cdot \,^{3}\!\!\!\sqrt{4}\\ \\ =2^{2}\cdot \,^{3}\!\!\!\sqrt{4}\\ \\ =4\,^{3}\!\!\!\sqrt{4}$


d) $\,^{4}\!\!\!\sqrt{1024}$

$=\,^{4}\!\!\!\sqrt{2^{10}}\\ \\ =\,^{4}\!\!\!\sqrt{2^{8+2}}\\ \\ =\,^{4}\!\!\!\sqrt{2^{8}\cdot 2^{2}}\\ \\ =\,^{4}\!\!\!\sqrt{2^{2\,\cdot\,4}\cdot 2^{2}}\\ \\ =\,^{4}\!\!\!\!\sqrt{(2^{2})^{4}\cdot 2^{2}}\\ \\ =\,^{4}\!\!\!\!\sqrt{(2^{2})^{4}\cdot 4}\\ \\ =\,^{4}\!\!\!\!\sqrt{(2^{2})^{4}}\cdot \,^{4}\!\!\!\sqrt{4}\\ \\ =2^{2}\cdot \,^{4}\!\!\!\sqrt{4}\\ \\ =4\,^{4}\!\!\!\sqrt{4}$


e) $\,^{4}\!\!\!\sqrt{7^{5}\cdot 7^{3}}$

$=\,^{4}\!\!\!\sqrt{7^{5+3}}\\ \\ =\,^{4}\!\!\!\sqrt{7^{8}}\\ \\ =\,^{4}\!\!\!\sqrt{7^{2\,\cdot\,4}}\\ \\ =\,^{4}\!\!\!\!\sqrt{(7^{2})^{4}}\\ \\ =7^{2}\\ \\ =49$


f) $\,^{10}\!\!\!\!\sqrt{2^{15}\cdot 3^{8}}$

$=\,^{10}\!\!\!\!\sqrt{2^{10+5}\cdot 3^{8}}\\ \\ =\,^{10}\!\!\!\!\sqrt{2^{10}\cdot 2^{5}\cdot 3^{8}}\\ \\ =\,^{10}\!\!\!\!\sqrt{2^{10}}\cdot \,^{10}\!\!\!\!\sqrt{2^{5}\cdot 3^{8}}\\ \\ =2\,^{10}\!\!\!\!\sqrt{2^{5}\cdot 3^{8}}$