因数 ((3^22^4)^5(12^518^{30})^4)^{-2}

解

因数

((3^{2} 2^{4})^{5} (1 2^{5} 1 8^{30})^{4})^{- 2}

\frac{1}{3 ^{540} \cdot 2 ^{360}}

解答ステップ

((3^{2} \cdot 2^{4})^{5} (1 2^{5} \cdot 1 8^{30})^{4})^{- 2}

(3^{2} \cdot 2^{4})^{5} = 2^{20} \cdot 3^{10}

= (2^{20} \cdot 3^{10} (1 8^{30} \cdot 1 2^{5})^{4})^{- 2}

(1 2^{5} \cdot 1 8^{30})^{4} = 3^{260} \cdot 2^{160}

= (3^{260} \cdot 2^{160} \cdot 2^{20} \cdot 3^{10})^{- 2}

簡素化

3^{10} \cdot 2^{20} \cdot 2^{160} \cdot 3^{260} : 3^{270} \cdot 2^{180}

= (3^{270} \cdot 2^{180})^{- 2}

指数の規則を適用する:

a^{- b} = \frac{1}{a ^{b}}

= \frac{1}{( 3 ^{270} \cdot 2 ^{180} ) ^{2}}

指数の規則を適用する:

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

(3^{270} \cdot 2^{180})^{2} = (2^{180})^{2} (3^{270})^{2}

= \frac{1}{( 2 ^{180} ) ^{2} ( 3 ^{270} ) ^{2}}

(2^{180})^{2} : 2^{360}

= \frac{1}{2 ^{360} ( 3 ^{270} ) ^{2}}

(3^{270})^{2} : 3^{540}

= \frac{1}{2 ^{360} \cdot 3 ^{540}}