展開する (t^{1/2}x^{1/2}+t^{1/3}y^{1/3})^3

解

展開する

(t^{\frac{1}{2}} x^{\frac{1}{2}} + t^{\frac{1}{3}} y^{\frac{1}{3}})^{3}

t^{\frac{3}{2}} x^{\frac{3}{2}} + 3 t^{\frac{4}{3}} x y^{\frac{1}{3}} + 3 t^{\frac{7}{6}} x^{\frac{1}{2}} y^{\frac{2}{3}} + t y

解答ステップ

(t^{\frac{1}{2}} x^{\frac{1}{2}} + t^{\frac{1}{3}} y^{\frac{1}{3}})^{3}

完全立方式を適用する:

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

(t^{\frac{1}{2}} x^{\frac{1}{2}} + t^{\frac{1}{3}} y^{\frac{1}{3}})^{3} = (t^{\frac{1}{2}} x^{\frac{1}{2}})^{3} + 3 (t^{\frac{1}{2}} x^{\frac{1}{2}})^{2} t^{\frac{1}{3}} y^{\frac{1}{3}} + 3 t^{\frac{1}{2}} x^{\frac{1}{2}} (t^{\frac{1}{3}} y^{\frac{1}{3}})^{2} + (t^{\frac{1}{3}} y^{\frac{1}{3}})^{3}

= (t^{\frac{1}{2}} x^{\frac{1}{2}})^{3} + 3 (t^{\frac{1}{2}} x^{\frac{1}{2}})^{2} t^{\frac{1}{3}} y^{\frac{1}{3}} + 3 t^{\frac{1}{2}} x^{\frac{1}{2}} (t^{\frac{1}{3}} y^{\frac{1}{3}})^{2} + (t^{\frac{1}{3}} y^{\frac{1}{3}})^{3}

(t^{\frac{1}{2}} x^{\frac{1}{2}})^{3} = t^{\frac{3}{2}} x^{\frac{3}{2}}

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

3 t^{\frac{1}{2}} x^{\frac{1}{2}} (t^{\frac{1}{3}} y^{\frac{1}{3}})^{2} = 3 t^{\frac{7}{6}} x^{\frac{1}{2}} y^{\frac{2}{3}}

(t^{\frac{1}{3}} y^{\frac{1}{3}})^{3} = t y

= t^{\frac{3}{2}} x^{\frac{3}{2}} + 3 t^{\frac{4}{3}} x y^{\frac{1}{3}} + 3 t^{\frac{7}{6}} x^{\frac{1}{2}} y^{\frac{2}{3}} + t y