化简 100(0.5^{1/32})^{39}

解答

化简

100 (0. 5^{\frac{1}{32}})^{39}

2^{\frac{25}{32}} \cdot 25

十进制

42.96548 \dots

求解步骤

100 (0. 5^{\frac{1}{32}})^{39}

使用指数法则:

(a^{b})^{c} = a^{b c}, a \geq 0

(0. 5^{\frac{1}{32}})^{39} = 0. 5^{\frac{1}{32} \cdot 39}

= 100 \cdot 0. 5^{\frac{1}{32} \cdot 39}

\frac{1}{32} \cdot 39 = \frac{39}{32}

= 100 \cdot 0. 5^{\frac{39}{32}}

0.5 = \frac{1}{2}

= 100 (\frac{1}{2})^{\frac{39}{32}}

使用指数法则:

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

(\frac{1}{2})^{\frac{39}{32}} = \frac{1 ^{\frac{39}{32}}}{2 ^{\frac{39}{32}}}

= 100 \cdot \frac{1 ^{\frac{39}{32}}}{2 ^{\frac{39}{32}}}

\frac{1 ^{\frac{39}{32}}}{2 ^{\frac{39}{32}}} = \frac{1}{2 \cdot 2 ^{\frac{7}{32}}}

= 100 \cdot \frac{1}{2 \cdot 2 ^{\frac{7}{32}}}

100 \cdot \frac{1}{2 \cdot 2 ^{\frac{7}{32}}} = 2^{\frac{25}{32}} \cdot 25

= 2^{\frac{25}{32}} \cdot 25

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