Question

Escreva a forma de potência mais simples do radical abaixo:
ALG ME AJUDA POR FAVOR​

Answer 1

$\sqrt[3]{11^{29}\sqrt[4]{11^{28}\sqrt[5]{11^{27}\sqrt[6]{11^{26}\sqrt[7]{11^{25}}}}}}$

Utilizando a seguinte propriedade de potencia:

$\sqrt[a]{b^c}=b^{\frac{c}{a}}$

Podemos escrever esse radical da seguinte forma

$(11^{29}(11^{28}(11^{27}(11^{26}\cdot11^\frac{27}{5})^{\frac{1}{6}})^\frac{1}{5})^\frac{1}{4})^\frac{1}{3}$

Distribuindo os  expoentes e respeitando a ordem dos parenteses chegamos a

$11^{\frac{29}{3}}\cdot11^{\frac{28}{12}}\cdot11^{\frac{27}{60}}\cdot11^{\frac{26}{360}}\cdot11^{\frac{25}{2520}}$

$11^{\frac{29}{3}+{\frac{28}{12}}+{\frac{27}{60}}+{\frac{26}{360}}+{\frac{25}{2520}}$

fiz o calculo da frações no expoente utilizando a calculadora do geogebra, então segue em anexo o resultado, por fim

$\sqrt[3]{11^{29}\sqrt[4]{11^{28}\sqrt[5]{11^{27}\sqrt[6]{11^{26}\sqrt[7]{11^{25}}}}}}=11^{\frac{29}{3}+\frac{28}{12}+\frac{27}{60}+\frac{26}{360}+\frac{25}{2520}}=\boxed{\boxed{11^{\frac{3509}{280}}}}$

Uma segunda solução seria fazer os numeros que estão fora dos radicais internos "voltarem" para o radical com o mesmo indice, desse jeito

$\sqrt[3]{11^{29}\sqrt[4]{11^{28}\sqrt[5]{11^{27}\sqrt[6]{11^{26}\sqrt[7]{11^{25}}}}}}$

$\sqrt[3]{\sqrt[4]{(11^{29})^4\cdot11^{28}\sqrt[5]{11^{27}\sqrt[6]{11^{26}\sqrt[7]{11^{25}}}}}}\\\\\sqrt[3]{\sqrt[4]{11^{116}\cdot11^{28}\sqrt[5]{11^{27}\sqrt[6]{11^{26}\sqrt[7]{11^{25}}}}}}\\\\\sqrt[3]{\sqrt[4]{11^{116+28}\sqrt[5]{11^{27}\sqrt[6]{11^{26}\sqrt[7]{11^{25}}}}}}\\\\\sqrt[3]{\sqrt[4]{11^{144}\sqrt[5]{11^{27}\sqrt[6]{11^{26}\sqrt[7]{11^{25}}}}}}\\\\\sqrt[3]{\sqrt[4]{\sqrt[5]{(11^{144})^511^{27}\sqrt[6]{11^{26}\sqrt[7]{11^{25}}}}}}$

$\sqrt[3]{\sqrt[4]{\sqrt[5]{(11^{144})^511^{27}\sqrt[6]{11^{26}\sqrt[7]{11^{25}}}}}}\\\\\sqrt[3]{\sqrt[4]{\sqrt[5]{11^{720}11^{27}\sqrt[6]{11^{26}\sqrt[7]{11^{25}}}}}}\\\\\sqrt[3]{\sqrt[4]{\sqrt[5]{11^{720+27}\sqrt[6]{11^{26}\sqrt[7]{11^{25}}}}}}\\\\\sqrt[3]{\sqrt[4]{\sqrt[5]{11^{747}\sqrt[6]{11^{26}\sqrt[7]{11^{25}}}}}}\\\\\sqrt[3]{\sqrt[4]{\sqrt[5]{\sqrt[6]{(11^{747})^611^{26}\sqrt[7]{11^{25}}}}}}\\\\\sqrt[3]{\sqrt[4]{\sqrt[5]{\sqrt[6]{11^{4482}11^{26}\sqrt[7]{11^{25}}}}}}$

$\sqrt[3]{\sqrt[4]{\sqrt[5]{\sqrt[6]{11^{4482+26}\sqrt[7]{11^{25}}}}}}\\\\\sqrt[3]{\sqrt[4]{\sqrt[5]{\sqrt[6]{11^{4508}\sqrt[7]{11^{25}}}}}}\\\\\sqrt[3]{\sqrt[4]{\sqrt[5]{\sqrt[6]{\sqrt[7]{(11^{4508})^711^{25}}}}}}\\\\\sqrt[3]{\sqrt[4]{\sqrt[5]{\sqrt[6]{\sqrt[7]{11^{31556}11^{25}}}}}}\\\\\sqrt[3]{\sqrt[4]{\sqrt[5]{\sqrt[6]{\sqrt[7]{11^{31556+25}}}}}}\\\\\sqrt[3]{\sqrt[4]{\sqrt[5]{\sqrt[6]{\sqrt[7]{11^{31581}}}}}}$

Utilizando a seguinte propriedade

$\sqrt[a]{\sqrt[b]{a}}=\sqrt[a\cdot b]{a}$

$\sqrt[3\cdot4\cdot5\cdot6\cdot7]{11^{31581}}\\\\11^{\frac{31581}{2520}}$, simplificando numerador e denominador por 9 chegamos a

$\boxed{\boxed{11^{\frac{3509}{280}}}}$