Question

reduza a um único radical e em seguida simplique se possível.
a)
$\sqrt{ \sqrt{10} }$
b)
$3 \sqrt{ \sqrt{3} }$
c)
$\sqrt{ \sqrt{ \sqrt{2} } }$
d)
$3 \sqrt{3 \sqrt{ \sqrt{3} } }$


mim ajudem por favor
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Answer 1

Resposta:

a) $\sqrt[4]{10}$

b) $3\sqrt[4]{3}$

c) $\sqrt[8]{2}$

d) $3\cdot \:3^{\frac{5}{8}}$

Explicação passo a passo:

a) $\sqrt{\sqrt{10}}$

$=\left(10^{\frac{1}{2}}\right)^{\frac{1}{2}}$

$=10^{\frac{1}{2}\cdot \frac{1}{2}}$

$=10^{\frac{1}{4}}$

$=\sqrt[4]{10}$

b) $3\sqrt{\sqrt{3}}$

$=3\left(3^{\frac{1}{2}}\right)^{\frac{1}{2}}$

$=3\left(3^{\frac{1}{2}\cdot \:\frac{1}{2}}\right)$

$=3\left(3^{\frac{1}{4}}\right)$

$=3\sqrt[4]{3}$

c) $\sqrt{\sqrt{\sqrt{2}}}$

$=\sqrt{\sqrt[4]{2}}$

$=\left(2^{\frac{1}{4}}\right)^{\frac{1}{2}}$

$=2^{\frac{1}{4}\cdot \frac{1}{2}}$

$=2^{\frac{1}{8}}$

$=\sqrt[8]{2}$

d) $3\sqrt{3\sqrt{\sqrt{3}}}$

$=3\sqrt{3\sqrt[4]{3}}$

$=3\left(\sqrt{3}\sqrt{\sqrt[4]{3}}\right)$

$=3\left(\sqrt{3}\sqrt[8]{3}\right)$

$=3\left(3^{\frac{1}{2}+\frac{1}{8}}\right)$

$=3\left(3^{\frac{5}{8}}\right)$

$=3\cdot \:3^{\frac{5}{8}}$