Question

Expresse as Potências Como Radicais e os Radicais Como Potências.

CÁLCULOS.

Letra A & E já foram respondidas.

A)

$\sqrt{ \sqrt{ {5}^{2} } } = \sqrt[4]{25} = {25}^{ \frac{1}{4} }$

B)

$(3 {}^{2}) {}^{ \frac{1}{3} }$

C)

$\sqrt[5]{ \sqrt{3} \times \sqrt[4]{2} }$

D)

$3 \sqrt[3]{2 \sqrt[3]{2 \sqrt{2} } }$
E)

$\sqrt[3]{ \sqrt{18} } = \sqrt[6]{18} = {18}^{ \frac{1}{6} }$

F)

$(2 {}^{ \frac{1}{5} }) {}^{ \frac{2}{3} }$

G)

$\sqrt{ \sqrt{ \sqrt{30} } }$

H)

$\sqrt{2 \sqrt{2 \sqrt{2} } }$

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Answer 1

Resposta:

Explicação passo a passo:

a) Respondida

b)

(3²)¹/³=∛3²=∛9

c)

$\displaystyle \sqrt[5]{\sqrt{3}\times\sqrt[4]{2}} =\sqrt[5]{\sqrt{3}}\times\sqrt[5]{\sqrt[4]{2}} =\sqrt[(5\times2)]{3}\times\sqrt[(5\times4)]{2} =\sqrt[10]{3} \times\sqrt[20]{2}=3^{\frac{1}{10}}\times2^{\frac{1}{20}}$

d)

$\displaystyle 3\sqrt[3]{2\sqrt[3]{2\sqrt{2}}}=3\sqrt[3]{2\sqrt[3]{2.2^{\frac{1}{2} }}}=3\sqrt[3]{2\sqrt[3]{2^{(1+\frac{1}{2})}}}=3\sqrt[3]{2\sqrt[3]{2^{\frac{3}{2}}}}=3\sqrt[3]{2.{2^{(\frac{3}{2})(\frac{1}{3})}}}=3\sqrt[3]{2.{2^{\frac{1}{2}}}}=3\sqrt[3]{{2^{(1+\frac{1}{2})}}}=3\sqrt[3]{{2^{\frac{3}{2}}}}=3.{{2^{(\frac{3}{2})(\frac{1}{3})}}}=3.2^{\frac{1}{2} }$

e) Respondida

f)

$\displastyle (2^\frac{^1}{5})^\frac{2}{3} =2^{\frac{2.1}{5.3}}=2^{\frac{2}{15}}=\sqrt[15]{2^2} =\sqrt[15]{4}$

g)

$\displaystyle \sqrt{\sqrt{\sqrt{30}}} =\sqrt{\sqrt{30^\frac{1}{2}}} =\sqrt{30^{(\frac{1}{2} )(\frac{1}{2} )}} =30^{(\frac{1}{2} )(\frac{1}{2} )(\frac{1}{2} )}=30^{(\frac{1.1.1}{2.2.2})}=30^\frac{1}{8}$

h)

$\displaystyle \sqrt{2\sqrt{2\sqrt{2}}} =\sqrt{2\sqrt{2.2^\frac{1}{2} }} =\sqrt{2\sqrt{2^\frac{3}{2}}} =\sqrt{2.{2^{(\frac{3}{2})(\frac{1}{2} )}}}=\sqrt{2.2^\frac{3}{4}}=\sqrt{2^\frac{7}{4}}=2^{(\frac{7}{4} )(\frac{1}{2} )}=2^\frac{7}{8}$