化简 432^{1/4}

解答

化简

43 2^{\frac{1}{4}}

2 \cdot 3^{\frac{3}{4}}

十进制

4.55901 \dots

求解步骤

43 2^{\frac{1}{4}}

432

质因数分解:

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

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

使用指数法则:

(a \cdot b)^{n} = a^{n} b^{n}

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

= (2^{4})^{\frac{1}{4}} (3^{3})^{\frac{1}{4}}

(2^{4})^{\frac{1}{4}} = 2

= 2 (3^{3})^{\frac{1}{4}}

使用指数法则:

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

(3^{3})^{\frac{1}{4}} = 3^{3 \cdot \frac{1}{4}}

= 2 \cdot 3^{3 \cdot \frac{1}{4}}

3 \cdot \frac{1}{4} = \frac{3}{4}

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

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