簡約 ((K/L)^{1/2})/((L/K)^{1/2)}

解

簡約

\frac{( \frac{K}{L} ) ^{\frac{1}{2}}}{( \frac{L}{K} ) ^{\frac{1}{2}}}

\frac{K}{L}

解答ステップ

\frac{( \frac{K}{L} ) ^{\frac{1}{2}}}{( \frac{L}{K} ) ^{\frac{1}{2}}}

指数の規則を適用する:

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

\frac{( \frac{K}{L} ) ^{\frac{1}{2}}}{( \frac{L}{K} ) ^{\frac{1}{2}}} = (\frac{\frac{K}{L}}{\frac{L}{K}})^{\frac{1}{2}}

= (\frac{\frac{K}{L}}{\frac{L}{K}})^{\frac{1}{2}}

\frac{\frac{K}{L}}{\frac{L}{K}} = \frac{K ^{2}}{L ^{2}}

= (\frac{K ^{2}}{L ^{2}})^{\frac{1}{2}}

累乗根の規則を適用する:

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

, 以下を想定

a \geq 0, b \geq 0

= \frac{( K ^{2} ) ^{\frac{1}{2}}}{( L ^{2} ) ^{\frac{1}{2}}}

(L^{2})^{\frac{1}{2}} : L

= \frac{( K ^{2} ) ^{\frac{1}{2}}}{L}

(K^{2})^{\frac{1}{2}} : K

= \frac{K}{L}