Olá,
Temos as raízes:
01)
[tex] \tt \: \sqrt[4]{ \dfrac{144}{121} } [/tex]
Vamos encontrar os fatores primos de 144 e 121.
[tex] \tt \: 144 | 2 \\ \tt \: \: \: 72 | 2 \\ \tt \: 36 | 2 \\ \tt18 | 2 \\ \tt9 | 3 \\ \tt3 | 3 \\ \tt1[/tex]
[tex] \tt \: 121 | 11 \\ \tt11 | 11 \\ \tt1[/tex]
Temos, então:
[tex] \tt \: 144 = {2}^{4} \cdot {3}^{2} \\ \\ \tt \: 121 = {11}^{2} [/tex]
Assim:
[tex] \tt \: \sqrt[4]{ \dfrac{144}{121} } \\ \\ \tt= \frac{ \sqrt[4]{144} }{ \sqrt[4]{121} } \\ \\ \tt = \frac{ \sqrt[4]{ {2}^{4} \cdot {3}^{2} } }{ \sqrt[4]{ {11}^{2} } } \\ \\ \tt = \frac{ \sqrt[4]{ {2}^{4} } \cdot \sqrt[4]{ {3}^{2} } }{ \sqrt[4]{ {11}^{2} } } \\ \\ \tt = \frac{ \sqrt[ \cancel{4}]{ {2}^{ \cancel{4} }} \cdot \sqrt[ \frac{4}{2} ]{ {3}^{ \frac{2}{2} } } }{ \sqrt[ \frac{4}{2} ]{ {11}^{ \frac{4}{2} } } } \\ \\ \tt = \dfrac{2 \sqrt{3} }{ \sqrt{11} } [/tex]
Resposta:
[tex] \boxed{ \tt \: \sqrt[4]{ \dfrac{144}{121} } = \frac{2 \sqrt{3} }{ \sqrt{11} } } \\ [/tex]
02)
[tex] \tt \sqrt[5]{ \dfrac{1024}{32} } [/tex]
Fatorando:
[tex] \tt \: 1024 | 2 \\ \tt \: 512 | 2 \\ \tt256 | 2 \\ \tt128 | 2 \\ \tt64 | 2 \\ \tt32 |2 \\ \tt 16 | 2 \\ \tt8 | 2 \\ \tt4 | 2 \\ \tt2 | 2 \\ \tt1[/tex]
[tex] \tt32 | 2 \\ \tt16 | 2 \\ \tt8 | 2 \\ \tt4 | 2 \\ \tt2 | 2 \\ \tt1[/tex]
Temos:
[tex] \tt1024 = {2}^{10} \\ \\ \tt32 = {2}^{5} [/tex]
Logo:
[tex] \tt \sqrt[5]{ \dfrac{1024}{32} } \\ \\ \tt = \tt \sqrt[5]{ \dfrac{ {2}^{10} }{ {2}^{5} } } \\ \\ \tt = \sqrt[5]{ {2}^{10 - 5} } \\ \\ \tt = \sqrt[5]{ {2}^{5} } \\ \\ \tt = \sqrt[ \cancel5]{ {2}^{ \cancel5} } \\ \\ \tt = 2[/tex]
Resposta:
[tex] \boxed{\tt \sqrt[5]{ \dfrac{1024}{32} } = 2} \\ [/tex]
