簡約 [(5/3)^{-3}(5/3)^{-2}(5/3)^5]^{12}

解

簡約

[(\frac{5}{3})^{- 3} (\frac{5}{3})^{- 2} (\frac{5}{3})^{5}]^{12}

1

解答ステップ

((\frac{5}{3})^{- 3} (\frac{5}{3})^{- 2} (\frac{5}{3})^{5})^{12}

指数の規則を適用する:

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

((\frac{5}{3})^{- 3} (\frac{5}{3})^{- 2} (\frac{5}{3})^{5})^{12} = ((\frac{5}{3})^{- 3})^{12} ((\frac{5}{3})^{- 2})^{12} ((\frac{5}{3})^{5})^{12}

= ((\frac{5}{3})^{- 3})^{12} ((\frac{5}{3})^{- 2})^{12} ((\frac{5}{3})^{5})^{12}

((\frac{5}{3})^{- 3})^{12} = \frac{3 ^{36}}{5 ^{36}}

= \frac{3 ^{36}}{5 ^{36}} ((\frac{5}{3})^{- 2})^{12} ((\frac{5}{3})^{5})^{12}

((\frac{5}{3})^{- 2})^{12} = \frac{3 ^{24}}{5 ^{24}}

= \frac{3 ^{36}}{5 ^{36}} \cdot \frac{3 ^{24}}{5 ^{24}} ((\frac{5}{3})^{5})^{12}

((\frac{5}{3})^{5})^{12} = \frac{5 ^{60}}{3 ^{60}}

= \frac{3 ^{36}}{5 ^{36}} \cdot \frac{3 ^{24}}{5 ^{24}} \cdot \frac{5 ^{60}}{3 ^{60}}

分数を乗じる:

\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}

= \frac{3 ^{36} \cdot 3 ^{24} \cdot 5 ^{60}}{5 ^{36} \cdot 5 ^{24} \cdot 3 ^{60}}

3^{36} \cdot 3^{24} \cdot 5^{60} = 3^{60} \cdot 5^{60}

= \frac{3 ^{60} \cdot 5 ^{60}}{3 ^{60} \cdot 5 ^{36} \cdot 5 ^{24}}

5^{36} \cdot 5^{24} \cdot 3^{60} = 5^{60} \cdot 3^{60}

= \frac{3 ^{60} \cdot 5 ^{60}}{5 ^{60} \cdot 3 ^{60}}

共通因数を約分する:

3^{60} \cdot 5^{60}

= 1