Question

Preciso de ajuda
urgente

Answer 1

Resposta:

Explicação passo-a-passo:

a)

$\displaystyle \sqrt[12]{\frac{64}{729} }$

Fatorar 64:

64 | 2

32 | 2

16 | 2

8 | 2

4 | 2

2 | 2

1 | 1

64= 2 . 2 . 2 . 2 . 2 . 2 = 2⁶

Fatorar 729:

729 | 3

243 | 3

81 | 3

27 | 3

9 | 3

3 | 3

1 | 1

729 = 3 . 3 . 3 . 3 . 3 . 3 = 3⁶

$\displaystyle \sqrt[12]{\frac{64}{729} }=\sqrt[12]{\frac{2^6}{3^6} }=\sqrt[12]{(\frac{2}{3})^6}=(\frac{2}{3})^{\frac{6}{12} }=(\frac{2}{3})^{\frac{1}{2} }=\sqrt{\frac{2}{3}}$

b)

$\displaystyle \sqrt[4]{\frac{144}{121} }$

Fatorar 144:

144 | 2

72 | 2

36 | 2

18 | 2

9 | 3

3 | 3

1 | 1

144 = 2 . 2 . 2 . 2 . 3 . 3 = 2⁴.3²

Fatorar 121

121 | 11

11 | 11

1 | 1

121 = 11 . 11 = 11²

$\displaystyle \sqrt[4]{\frac{144}{121} }=\sqrt[4]{\frac{2^4.3^2}{11^2} }=\frac{2^{\frac{4}{4} }.3^{\frac{2}{4} }}{11^{\frac{2}{4} }}=\frac{2.3^{\frac{1}{2} }}{11^{\frac{1}{2} }}= 2.\frac{\sqrt{3} }{\sqrt{11} } .\frac{\sqrt{11}}{\sqrt{11}} =\frac{2\sqrt{33} }{11}$

c)

$\displaystyle \sqrt[5]{\frac{1024}{32} }$

Fatorar 1024:

1024 | 2

512 | 2

256 | 2

128 | 2

64 | 2

32 | 2

16 | 2

8 | 2

4 | 2

2 | 2

1 | 1

1024 = 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 = 2¹⁰

Fatorar 32:

32 | 2

16 | 2

8 | 2

4 | 2

2 | 2

1 | 1

32=2.2.2.2.2=2⁵

$\displaystyle \sqrt[5]{\frac{1024}{32} }= \sqrt[5]{\frac{2^{10}}{2^{5}} }=\sqrt[5]{2^{(10-5)}}= \sqrt[5]{2^{5}}= 2^{\frac{5}{5} }=2^1=2$