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simplificar (12)^{-1}(144)^2(12)^{-3}*(1728)^4

Solução

simplificar

(12)^{- 1} \cdot (144)^{2} \cdot (12)^{- 3} \cdot (1728)^{4}

8916100448256

Passos da solução

1 2^{- 1} \cdot 14 4^{2} \cdot 1 2^{- 3} \cdot 172 8^{4}

1 2^{- 1} \cdot 1 2^{- 3} = \frac{1}{20736}

= \frac{1}{20736} \cdot 14 4^{2} \cdot 172 8^{4}

Fatorar o inteiro

144 = 2^{4} \cdot 3^{2}

= \frac{1}{20736} (2^{4} \cdot 3^{2})^{2} \cdot 172 8^{4}

Aplicar as propriedades dos expoentes:

(ab)^{c} = a^{c} b^{c}

(2^{4} \cdot 3^{2})^{2} = (2^{4})^{2} (3^{2})^{2}

= \frac{1}{20736} (2^{4})^{2} (3^{2})^{2} \cdot 172 8^{4}

Aplicar as propriedades dos expoentes:

(a^{b})^{c} = a^{b c}

(2^{4})^{2} = 2^{4 \cdot 2}, (3^{2})^{2} = 3^{2 \cdot 2}

= \frac{1}{20736} \cdot 2^{4 \cdot 2} \cdot 3^{2 \cdot 2} \cdot 172 8^{4}

Simplificar

= \frac{1}{20736} \cdot 2^{8} \cdot 3^{4} \cdot 172 8^{4}

Fatorar o inteiro

1728 = 2^{6} \cdot 3^{3}

= \frac{1}{20736} \cdot 2^{8} \cdot 3^{4} (2^{6} \cdot 3^{3})^{4}

Aplicar as propriedades dos expoentes:

(ab)^{c} = a^{c} b^{c}

(2^{6} \cdot 3^{3})^{4} = (2^{6})^{4} (3^{3})^{4}

= \frac{1}{20736} \cdot 2^{8} \cdot 3^{4} (2^{6})^{4} (3^{3})^{4}

Aplicar as propriedades dos expoentes:

(a^{b})^{c} = a^{b c}

(2^{6})^{4} = 2^{6 \cdot 4}, (3^{3})^{4} = 3^{3 \cdot 4}

= \frac{1}{20736} \cdot 2^{8} \cdot 3^{4} \cdot 2^{6 \cdot 4} \cdot 3^{3 \cdot 4}

Simplificar

= \frac{1}{20736} \cdot 2^{8} \cdot 3^{4} \cdot 2^{24} \cdot 3^{12}

Aplicar as propriedades dos expoentes:

a^{b} \cdot a^{c} = a^{b + c}

2^{8} \cdot 2^{24} = 2^{8 + 24}

= \frac{1}{20736} \cdot 3^{4} \cdot 2^{8 + 24} \cdot 3^{12}

Somar:

8 + 24 = 32

= \frac{1}{20736} \cdot 3^{4} \cdot 2^{32} \cdot 3^{12}

Aplicar as propriedades dos expoentes:

a^{b} \cdot a^{c} = a^{b + c}

3^{4} \cdot 3^{12} = 3^{4 + 12}

= \frac{1}{20736} \cdot 2^{32} \cdot 3^{4 + 12}

Somar:

4 + 12 = 16

= \frac{1}{20736} \cdot 2^{32} \cdot 3^{16}

Multiplicar frações:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2 ^{32} \cdot 3 ^{16}}{20736}

Multiplicar:

1 \cdot 2^{32} = 2^{32}

= \frac{2 ^{32} \cdot 3 ^{16}}{20736}

Fatorar

20736 : 2^{8} \cdot 3^{4}

= \frac{2 ^{32} \cdot 3 ^{16}}{2 ^{8} \cdot 3 ^{4}}

Cancelar

\frac{2 ^{32} \cdot 3 ^{16}}{2 ^{8} \cdot 3 ^{4}} : 2^{24} \cdot 3^{12}

= 2^{24} \cdot 3^{12}

2^{24} = 16777216

= 3^{12} \cdot 16777216

3^{12} = 531441

= 16777216 \cdot 531441

Multiplicar os números:

16777216 \cdot 531441 = 8916100448256

= 8916100448256

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