Radicais

por RamonLucas Sáb 15 Jul 2017, 11:03

A expressão $\sqrt{a^{2}+\sqrt[3]{a^{4}b^{2}}}+\sqrt{b^{2}+\sqrt[3]{a^{2}b^{4}}}$ é igual a:

a) $\left ( a^{\frac{1}{3}} +b^{\frac{1}{3}}\right )^{\frac{3}{2}}$

b) $\left ( a^{\frac{3}{2}} +b^{\frac{3}{2}}\right )^{\frac{2}{3}}$

c) $\left ( a^{\frac{2}{3}} +b^{\frac{2}{3}}\right )^{\frac{3}{2}}$

d) $\left ( a+b \right )^{\frac{^3}{2}}$

e) $\left ( a^{\frac{3}{2}} +b^{\frac{3}{2}}\right )^{\frac{3}{2}}$

Não possuo gabarito

por Castiel Sáb 15 Jul 2017, 11:44

$\sqrt{a^2 + a^{4/3}.b^{2/3}} + \sqrt{b^2 + a^{2/3}.b^{4/3}}$

$\sqrt{a^{4/3}.(a^{2/3} + b^{2/3})} + \sqrt{b^{4/3}.(a^{2/3} + b^{2/3})}$

$a^{2/3}\sqrt{a^{2/3} + b^{2/3}} + b^{2/3}\sqrt{a^{2/3} + b^{2/3}}$

$(a^{2/3} + b^{2/3})(\sqrt{a^{2/3} + b^{2/3}})$

Igualando a expressão a x:

$(a^{2/3} + b^{2/3})(\sqrt{a^{2/3} + b^{2/3}}) = x$

$a^{2/3} + b^{2/3} = y$

$y.y^{1/2} = x$

$x = y^{3/2}$

$x = (a^{2/3} + b^{2/3})^{3/2}$

$(C)$

por RamonLucas Dom 16 Jul 2017, 15:37

Castiel escreveu: $\sqrt{a^2 + a^{4/3}.b^{2/3}} + \sqrt{b^2 + a^{2/3}.b^{4/3}}$

$\sqrt{a^{4/3}.(a^{2/3} + b^{2/3})} + \sqrt{b^{4/3}.(a^{2/3} + b^{2/3})}$

$a^{2/3}\sqrt{a^{2/3} + b^{2/3}} + b^{2/3}\sqrt{a^{2/3} + b^{2/3}}$

$(a^{2/3} + b^{2/3})(\sqrt{a^{2/3} + b^{2/3}})$

Igualando a expressão a x:

$(a^{2/3} + b^{2/3})(\sqrt{a^{2/3} + b^{2/3}}) = x$

$a^{2/3} + b^{2/3} = y$

$y.y^{1/2} = x$

$x = y^{3/2}$

$x = (a^{2/3} + b^{2/3})^{3/2}$

$(C)$

Obrigado.

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