簡約 (sqrt(\sqrt{ab^3)})^3

解

簡約

(a b^{3})^{3}

4 a^{3} 4 b^{9}

解答ステップ

(a b^{3})^{3}

指数の規則を適用する:

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

(a b^{3})^{2 + 1} = (a b^{3})^{2} a b^{3}

= (a b^{3})^{2} a b^{3}

指数の規則を適用する:

(n a)^{n} = a,

, 以下を想定

n i s o dd or a \geq 0

(a b^{3})^{2} = a b^{3}

= a b^{3} a b^{3}

a b^{3} = (a b^{3})^{\frac{1}{2}}

a b^{3} = (a b^{3})^{\frac{1}{4}}

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

指数の規則を適用する:

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

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

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

\frac{1}{2} + \frac{1}{4} = \frac{3}{4}

= (a b^{3})^{\frac{3}{4}}

指数の規則を適用する:

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

(a b^{3})^{\frac{3}{4}} = a^{\frac{3}{4}} (b^{3})^{\frac{3}{4}}

= a^{\frac{3}{4}} (b^{3})^{\frac{3}{4}}

簡素化

(b^{3})^{\frac{3}{4}} : b^{\frac{9}{4}}

= a^{\frac{3}{4}} b^{\frac{9}{4}}

a^{\frac{3}{4}} = 4 a^{3}

= 4 a^{3} 4 b^{9}

b^{\frac{9}{4}} = 4 b^{9}

= 4 a^{3} 4 b^{9}