Question

Resolva as seguintes racionalização:

a) 8\√2¹⁸

b) 16\³⁰√2⁴

c) 20\³⁰√30

d) 15\⁶√25

e) 20\₃√20

Answer 1

a) $\dfrac{8}{\,^{20}\!\!\!\sqrt{2^{18}}}$

Multiplicando o numerador e o denominador por $\,^{20}\!\!\!\sqrt{2^{2}},$ temos

$=\dfrac{8\cdot \,^{20}\!\!\!\sqrt{2^{2}}}{\,^{20}\!\!\!\sqrt{2^{18}}\cdot \,^{20}\!\!\!\sqrt{2^{2}}}\\ \\ \\ =\dfrac{8\cdot \,^{20}\!\!\!\sqrt{2^{2}}}{\,^{20}\!\!\!\sqrt{2^{18}\cdot 2^{2}}}\\ \\ \\ =\dfrac{8\cdot \,^{20}\!\!\!\sqrt{2^{2}}}{\,^{20}\!\!\!\sqrt{2^{18+2}}}\\ \\ \\ =\dfrac{8\cdot \,^{20}\!\!\!\sqrt{2^{2}}}{\,^{20}\!\!\!\sqrt{2^{20}}}\\ \\ \\ =\dfrac{8\cdot \,^{20}\!\!\!\sqrt{2^{2}}}{2}\\ \\ \\ =\dfrac{\diagup\!\!\!\! 2\cdot 4\cdot \,^{20}\!\!\!\sqrt{2^{2}}}{\diagup\!\!\!\! 2}\\ \\ \\ =4\,^{20}\!\!\!\sqrt{2^{2}}\\ \\ \\ =4\,^{20}\!\!\!\sqrt{4}$


b) $\dfrac{16}{\,^{30}\!\!\!\sqrt{2^{4}}}$

Multiplicando o numerador e o denominador por $\,^{30}\!\!\!\sqrt{2^{26}}\,,$ temos

$=\dfrac{16\cdot \,^{30}\!\!\!\sqrt{2^{26}}}{\,^{30}\!\!\!\sqrt{2^{4}}\cdot \,^{30}\!\!\!\sqrt{2^{26}}}\\ \\ \\ =\dfrac{16\cdot \,^{30}\!\!\!\sqrt{2^{26}}}{\,^{30}\!\!\!\sqrt{2^{4}\cdot 2^{26}}}\\ \\ \\ =\dfrac{16\cdot \,^{30}\!\!\!\sqrt{2^{26}}}{\,^{30}\!\!\!\sqrt{2^{4+26}}}\\ \\ \\ =\dfrac{16\cdot \,^{30}\!\!\!\sqrt{2^{26}}}{\,^{30}\!\!\!\sqrt{2^{30}}}\\ \\ \\ =\dfrac{16\cdot \,^{30}\!\!\!\sqrt{2^{26}}}{2}\\ \\ \\ =\dfrac{\diagup\!\!\!\! 2\cdot 8\cdot \,^{30}\!\!\!\sqrt{2^{26}}}{\diagup\!\!\!\! 2}\\ \\ \\ =8\,^{30}\!\!\!\sqrt{2^{26}}$


c) $\dfrac{20}{\,^{30}\!\!\!\sqrt{30}}$

Multiplicando o numerador e o denominador por $\,^{30}\!\!\!\sqrt{30^{29}},$ temos

$=\dfrac{20\cdot \,^{30}\!\!\!\sqrt{30^{29}}}{\,^{30}\!\!\!\sqrt{30}\cdot \,^{30}\!\!\!\sqrt{30^{29}}}\\ \\ \\ =\dfrac{20\cdot \,^{30}\!\!\!\sqrt{30^{29}}}{\,^{30}\!\!\!\sqrt{30\cdot 30^{29}}}\\ \\ \\ =\dfrac{20\cdot \,^{30}\!\!\!\sqrt{30^{29}}}{\,^{30}\!\!\!\sqrt{30^{1+29}}}\\ \\ \\ =\dfrac{20\cdot \,^{30}\!\!\!\sqrt{30^{29}}}{\,^{30}\!\!\!\sqrt{30^{30}}}\\ \\ \\ =\dfrac{20\cdot \,^{30}\!\!\!\sqrt{30^{29}}}{30}\\ \\ \\ =\dfrac{\diagup\!\!\!\!\! 10\cdot 2\cdot \,^{30}\!\!\!\sqrt{30^{29}}}{\diagup\!\!\!\!\! 10\cdot 3}\\ \\ \\ =\dfrac{2\,^{30}\!\!\!\sqrt{30^{29}}}{3}$


d) $\dfrac{15}{\,^{6}\!\!\!\sqrt{25}}$

$=\dfrac{15}{\,^{6}\!\!\!\sqrt{5^{2}}}$


Multiplicando o numerador e o denominador por $\,^{6}\!\!\!\sqrt{5^{4}},$ temos

$=\dfrac{15\cdot \,^{6}\!\!\!\sqrt{5^{4}}}{\,^{6}\!\!\!\sqrt{5^{2}}\cdot \,^{6}\!\!\!\sqrt{5^{4}}}\\ \\ \\ =\dfrac{15\cdot \,^{6}\!\!\!\sqrt{5^{4}}}{\,^{6}\!\!\!\sqrt{5^{2}\cdot 5^{4}}}\\ \\ \\ =\dfrac{15\cdot \,^{6}\!\!\!\sqrt{5^{4}}}{\,^{6}\!\!\!\sqrt{5^{2+4}}}\\ \\ \\ =\dfrac{15\cdot \,^{6}\!\!\!\sqrt{5^{4}}}{\,^{6}\!\!\!\sqrt{5^{6}}}\\ \\ \\ =\dfrac{15\cdot \,^{6}\!\!\!\sqrt{5^{4}}}{5}\\ \\ \\ =\dfrac{\diagup\!\!\!\! 5\cdot 3\cdot \,^{6}\!\!\!\sqrt{5^{4}}}{\diagup\!\!\!\! 5}\\ \\ \\ =3\cdot \,^{6}\!\!\!\sqrt{5^{4}}\\ \\ \\ =3\,^{6}\!\!\!\sqrt{625}$


e) $\dfrac{20}{\,^{3}\!\!\!\sqrt{20}}$

$=\dfrac{20}{\,^{3}\!\!\!\sqrt{2^{2}\cdot 5}}$


Multiplicando o numerador e o denominador por $\,^{3}\!\!\!\sqrt{2\cdot 5^{2}},$ temos

$=\dfrac{20\cdot \,^{3}\!\!\!\sqrt{2\cdot 5^{2}}}{\,^{3}\!\!\!\sqrt{2^{2}\cdot 5}\cdot \,^{3}\!\!\!\sqrt{2\cdot 5^{2}}}\\ \\ \\ =\dfrac{20\cdot \,^{3}\!\!\!\sqrt{2\cdot 5^{2}}}{\,^{3}\!\!\!\sqrt{(2^{2}\cdot 5)\cdot (2\cdot 5^{2})}}\\ \\ \\ =\dfrac{20\cdot \,^{3}\!\!\!\sqrt{2\cdot 5^{2}}}{\,^{3}\!\!\!\sqrt{2^{2}\cdot 2\cdot 5\cdot 5^{2}}}\\ \\ \\ =\dfrac{20\cdot \,^{3}\!\!\!\sqrt{2\cdot 5^{2}}}{\,^{3}\!\!\!\sqrt{2^{2+1}\cdot 5^{1+2}}}\\ \\ \\ =\dfrac{20\cdot \,^{3}\!\!\!\sqrt{2\cdot 5^{2}}}{\,^{3}\!\!\!\sqrt{2^{3}\cdot 5^{3}}}\\ \\ \\ =\dfrac{20\cdot \,^{3}\!\!\!\sqrt{2\cdot 5^{2}}}{\,^{3}\!\!\!\!\sqrt{(2\cdot 5)^{3}}}$

$=\dfrac{20\cdot \,^{3}\!\!\!\sqrt{2\cdot 5^{2}}}{2\cdot 5}\\ \\ \\ =\dfrac{\diagup\!\!\!\!\! 10\cdot 2\cdot \,^{3}\!\!\!\sqrt{2\cdot 5^{2}}}{\diagup\!\!\!\!\! 10}\\ \\ \\ =2\cdot \,^{3}\!\!\!\sqrt{2\cdot 5^{2}}\\ \\ \\ =2\,^{3}\!\!\!\sqrt{50}$