展開する (x^{1/4}+y^{1/4})^4

解

展開する

(x^{\frac{1}{4}} + y^{\frac{1}{4}})^{4}

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

解答ステップ

(x^{\frac{1}{4}} + y^{\frac{1}{4}})^{4}

2項定理を適用する:

(a + b)^{n} = i = 0 \sum n (i n) a^{(n - i)} b^{i}

a = x^{\frac{1}{4}}, b = y^{\frac{1}{4}}

= i = 0 \sum 4 (i 4) (x^{\frac{1}{4}})^{(4 - i)} (y^{\frac{1}{4}})^{i}

総和を展開する

= \frac{4 !}{0 ! ( 4 - 0 ) !} (x^{\frac{1}{4}})^{4} (y^{\frac{1}{4}})^{0} + \frac{4 !}{1 ! ( 4 - 1 ) !} (x^{\frac{1}{4}})^{3} (y^{\frac{1}{4}})^{1} + \frac{4 !}{2 ! ( 4 - 2 ) !} (x^{\frac{1}{4}})^{2} (y^{\frac{1}{4}})^{2} + \frac{4 !}{3 ! ( 4 - 3 ) !} (x^{\frac{1}{4}})^{1} (y^{\frac{1}{4}})^{3} + \frac{4 !}{4 ! ( 4 - 4 ) !} (x^{\frac{1}{4}})^{0} (y^{\frac{1}{4}})^{4}

簡素化

\frac{4 !}{0 ! ( 4 - 0 ) !} (x^{\frac{1}{4}})^{4} (y^{\frac{1}{4}})^{0} : x

簡素化

\frac{4 !}{1 ! ( 4 - 1 ) !} (x^{\frac{1}{4}})^{3} (y^{\frac{1}{4}})^{1} : 4 x^{\frac{3}{4}} y^{\frac{1}{4}}

簡素化

\frac{4 !}{2 ! ( 4 - 2 ) !} (x^{\frac{1}{4}})^{2} (y^{\frac{1}{4}})^{2} : 6 x^{\frac{1}{2}} y^{\frac{1}{2}}

簡素化

\frac{4 !}{3 ! ( 4 - 3 ) !} (x^{\frac{1}{4}})^{1} (y^{\frac{1}{4}})^{3} : 4 x^{\frac{1}{4}} y^{\frac{3}{4}}

簡素化

\frac{4 !}{4 ! ( 4 - 4 ) !} (x^{\frac{1}{4}})^{0} (y^{\frac{1}{4}})^{4} : y

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