Question

(valendo 20 pontos)Simplifique a expressão abaixo⬇
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Answer 1

Vamos por partes, o primeiro monômio do numerador:

$8.9^{\frac{5}{20} }\\8.9^{\frac{1}{4} } \\8\sqrt[4]{9}$

Segundo monômio do numerador:

$2.6^{\frac{n}{2n} }\\2.6x^{\frac{1}{2} } \\2\sqrt{6}$

Primeiro monômio do denominador:

$\sqrt[6]{2^{3}.3}$

$\sqrt{2}.\sqrt[6]{3}$

Segundo monômio do denominador:

$4\sqrt[4]{2^4.3^2}\\4.2\sqrt{3}$

Junta tudo:

$\frac{8\sqrt[4]{9}+2\sqrt{6}}{\sqrt{2}.\sqrt[6]{3}+8\sqrt{3}}\\\\\frac{8\sqrt[4]{9}+2\sqrt{6}}{\sqrt{2}.\sqrt[6]{3}+8\sqrt{3}}.\frac{\sqrt{3}^{5}}{\sqrt{3}^{5}}\\ \\\frac{(8\sqrt[4]{9}+2\sqrt{6})\sqrt{3}^{5}  }{3\sqrt{2}+216}\\\\\frac{(8\sqrt[4]{3}\sqrt[4]{3} +2\sqrt{2}\sqrt{3})\sqrt{3}^{5}  }{3\sqrt{2}+216}\\\\\frac{24\sqrt[4]{3^{3}} +54\sqrt{2}}{3\sqrt{2}+216}$

Agora aplica a raiz décima no numerador e no denominador.

$\sqrt[10]{\frac{24\sqrt[4]{3^{3}} +54\sqrt{2}}{3\sqrt{2}+216}}\\\\\frac{\sqrt[10]{24\sqrt[4]{3^{3}} +54\sqrt{2}}}{\sqrt[10]{3\sqrt{2}+216}}\\\\\frac{\sqrt[10]{24\sqrt[4]{3^{3}} +54\sqrt{2}}}{\sqrt[10]{3\sqrt{2}+216}}.\frac{\sqrt[10]{3\sqrt{2}+216}^{9} }{\sqrt[10]{3\sqrt{2}+216 }^{9} } \\\\\frac{\sqrt[10]{24\sqrt[4]{3^{3}} +54\sqrt{2}(3\sqrt{2}+216)^9}}{3\sqrt{2}+216}}$

$\frac{\sqrt[10]{24\sqrt[4]{3^{3}} +54\sqrt{2}(3\sqrt{2}+216)^9}}{3\sqrt{2}+216}}\\\\(3\sqrt{2}+216)^9 = (3\sqrt{2} )^9 +9(3\sqrt{2})^8.216 + 36(3\sqrt{2} )^7.216^2+84(3\sqrt{2} )^6.216^3 + 126(3\sqrt{2})^5.216^4+126(3\sqrt{2})^4.216^5 + 84(3\sqrt{2})^3.216^6 + 36(3\sqrt{2})^2.216^7+9(3\sqrt{2}).216^8+216^9$

$314928\sqrt{2} + 204073344 + 29386561536\sqrt{2} + 4936942338048 + 266594886254592\sqrt{2} + 19194831810330624 + 460675963447934976\sqrt{2} + 14215144014964850688\\+255872592269367312384\sqrt{2} + 23490369077469200000\\\\256333534857088368640\sqrt{2} + 37724712861390790656$

$\frac{\sqrt[10]{24\sqrt[4]{3^{3}} +54\sqrt{2}(256333534857088368640\sqrt{2} + 37724712861390790656)}}{3\sqrt{2}+216}}\\\\\frac{\sqrt[10]{24\sqrt[4]{3^{3}} +54(512667069714176737280 + 37724712861390790656\sqrt{2} )}}{3\sqrt{2}+216}}$

Tá, chega, kkkkk, simplifiquei o máximo que pude desde acho 23:30 mais ou menos, tô há uma hora tentando e tenho que ir dormir, tentei simplificar o máximo que pude! Até! Mas foi um desafio e tanto!

Answer 2

Resposta:

$\sqrt[10]{ \frac{8 \sqrt[20]{ {9}^{5} } + 2 \sqrt[n]{6 \frac{n}{2} } }{ \sqrt[6]{24} + 4 \sqrt[4]{144} } } \\ \\ \sqrt[10]{ \frac{8 \sqrt[4]{9} + 2 \sqrt[n]{3n} }{ \sqrt[6]{24} + 8 \sqrt[4]{9} } } \\ \\ \sqrt[10]{ \frac{8 \sqrt[4]{ {3}^{2} } + 2 \sqrt[n]{3n} }{ \sqrt[6]{24} + 8 \sqrt[4]{ {3}^{2} } } } \\ \\ \sqrt[10]{ \frac{8 \sqrt{3} + 2 \sqrt[n]{3n} }{ \sqrt[6]{24} + 8 \sqrt{3} } } \\ \\ \frac{ \sqrt[10]{ 8\sqrt{3} + 2 \sqrt[n]{3n} } }{ \sqrt[10]{ \sqrt[6]{24} + 8 \sqrt{3} } } \\ \\ \frac{ \sqrt[10]{(8 \sqrt{3} + 2 \sqrt[n]{3n} ) \times (\sqrt[6]{24} + 8 \sqrt{3} {)}^{9} } \times (4 \sqrt[5]{1944} + 384 \sqrt{6} + 36864 \sqrt[6]{24} - 96 \sqrt[6]{3} - 3072 \sqrt[6]{243} - 294912 \sqrt{ 3}) }{24 - {192}^{3} }$

• Espero ter ajudado.