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simplificar \sqrt[10]{48}*\sqrt[20]{48^3}

Solução

simplificar

1048 \cdot 20 4 8^{3}

2103 \cdot 3^{\frac{3}{20}}

Decimal

2.63214 \dots

Passos da solução

1048 20 4 8^{3}

20 4 8^{3} : 4 8^{\frac{3}{20}}

= 1048 \cdot 4 8^{\frac{3}{20}}

Fatorar o inteiro

48 = 2^{4} \cdot 3

= 10 2^{4} \cdot 3 \cdot 4 8^{\frac{3}{20}}

Aplicar as propriedades dos radicais:

n ab = n a n b

10 2^{4} \cdot 3 = 10 2^{4} 103

= 10 2^{4} 103 \cdot 4 8^{\frac{3}{20}}

Aplicar as propriedades dos expoentes:

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

10 2^{4} = 2^{4 \cdot \frac{1}{10}}

= 2^{4 \cdot \frac{1}{10}} 103 \cdot 4 8^{\frac{3}{20}}

Simplificar

= 2^{\frac{2}{5}} 103 \cdot 4 8^{\frac{3}{20}}

Fatorar o inteiro

48 = 2^{4} \cdot 3

= 2^{\frac{2}{5}} 103 (2^{4} \cdot 3)^{\frac{3}{20}}

Aplicar as propriedades dos expoentes:

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

(2^{4} \cdot 3)^{\frac{3}{20}} = (2^{4})^{\frac{3}{20}} \cdot 3^{\frac{3}{20}}

= 2^{\frac{2}{5}} 103 (2^{4})^{\frac{3}{20}} \cdot 3^{\frac{3}{20}}

Aplicar as propriedades dos expoentes:

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

(2^{4})^{\frac{3}{20}} = 2^{4 \cdot \frac{3}{20}}

= 2^{\frac{2}{5}} 103 \cdot 2^{4 \cdot \frac{3}{20}} \cdot 3^{\frac{3}{20}}

Simplificar

= 2^{\frac{2}{5}} 103 \cdot 2^{\frac{3}{5}} \cdot 3^{\frac{3}{20}}

Aplicar as propriedades dos expoentes:

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

2^{\frac{2}{5}} \cdot 2^{\frac{3}{5}} = 2^{\frac{2}{5} + \frac{3}{5}}

= 103 \cdot 2^{\frac{2}{5} + \frac{3}{5}} \cdot 3^{\frac{3}{20}}

2^{\frac{2}{5} + \frac{3}{5}} = 2

= 2103 \cdot 3^{\frac{3}{20}}

s im pl i f y (2.8 \cdot 1 0^{4}) \cdot (3 \cdot 1 0^{2})

s im pl i f y 2.077 \cdot 1 0^{- 4}

s im pl i f y (f^{- 1} \circ f) (x)

s im pl i f y 2^{2^{2^{2}}}

s im pl i f y 1.02 (lo g_{10} (1228000))^{- 2.5}