Explicación paso a paso:
210)
[tex] \sqrt{-81} = \sqrt{81(-1)} = \sqrt{81} \cdot \overbrace{\sqrt{-1}}^{i} = 9i [/tex]
211)
[tex] 7\sqrt{-4} = 7\sqrt{4(-1)} = 7\sqrt{4} \cdot \overbrace{\sqrt{-1}}^{i} = 7 \cdot 2 \cdot i = 14i [/tex]
212)
[tex] \frac{2}{5}\sqrt{-144} = \frac{2}{5}\sqrt{144(-1)} = \frac{2}{5}\sqrt{144} \cdot \overbrace{\sqrt{-1}}^{i} = \frac{2}{5} \cdot 12 \cdot i = \frac{24}{5} i [/tex]
213)
[tex] -3\sqrt{-96} = -3\sqrt{96(-1)} = -3\sqrt{96} \cdot \overbrace{\sqrt{-1}}^{i} = -3 \sqrt{96} i [/tex]
Descomponiendo el 96 en factores primos:
$\sqrt{96} = \sqrt{2⁵ • 3} = \sqrt{2⁴ • 2 • 3} = \sqrt{2⁴} \sqrt{2 • 3} = 2^\frac{4}{2} \sqrt{6}= 2² \sqrt{6} = 4 \sqrt{6}$
Sustituyendo este valor:
[tex] -3\sqrt{96} i = -3 \cdot 4\sqrt{6} i = -12 \sqrt{6} i [/tex]
214)
[tex] i^{5} + i^{-5} = i^{5} + \frac{1}{i^{5}} = \frac{i^{10}+1}{i^{5}} = \frac{(i^{2})^{5}+1}{i^{4} \cdot i^{1}} = \frac{(-1)^{5}+1}{i^{2} \cdot i^{2} \cdot i} = \frac{-1+1}{-1 \cdot -1 \cdot i} = \frac{0}{i} = 0 [/tex]
215)
No se ve bien.
216)
[tex] 4(i^{11})^{3} - \frac{3}{i} = 4(i^{33}) - \frac{3}{i} = \frac{4(i^{34}) - 3}{i} = \frac{4(i^{2})^{17} - 3}{i} = \frac{4(-1)^{17} - 3}{i} = \frac{4(-1) - 3}{i} = \frac{-4 - 3}{i} = \frac{-7}{i} [/tex]
Haciendo la división con números complejos:
[tex] \frac{-7}{i} = \frac{-7}{i} \cdot \frac{i}{i} = \frac{-7i}{i²} = \frac{-7i}{-1} = 7i [/tex]
217)
[tex] 3i^{5} + 6i^{9} - 9i^{12} [/tex]
