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人気のある代数 >

展開する (x^{1/3}+y^{1/3})^{12}

解

展開する

(x^{\frac{1}{3}} + y^{\frac{1}{3}})^{12}

解

x^{4} + 12 x^{\frac{11}{3}} y^{\frac{1}{3}} + 66 x^{\frac{10}{3}} y^{\frac{2}{3}} + 220 x^{3} y + 495 x^{\frac{8}{3}} y^{\frac{4}{3}} + 792 x^{\frac{7}{3}} y^{\frac{5}{3}} + 924 x^{2} y^{2} + 792 x^{\frac{5}{3}} y^{\frac{7}{3}} + 495 x^{\frac{4}{3}} y^{\frac{8}{3}} + 220 x y^{3} + 66 x^{\frac{2}{3}} y^{\frac{10}{3}} + 12 x^{\frac{1}{3}} y^{\frac{11}{3}} + y^{4}

解答ステップ

(x^{\frac{1}{3}} + y^{\frac{1}{3}})^{12}

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}{3}}, b = y^{\frac{1}{3}}

= i = 0 \sum 12 (i 12) (x^{\frac{1}{3}})^{(12 - i)} (y^{\frac{1}{3}})^{i}

総和を展開する

= \frac{12 !}{0 ! ( 12 - 0 ) !} (x^{\frac{1}{3}})^{12} (y^{\frac{1}{3}})^{0} + \frac{12 !}{1 ! ( 12 - 1 ) !} (x^{\frac{1}{3}})^{11} (y^{\frac{1}{3}})^{1} + \frac{12 !}{2 ! ( 12 - 2 ) !} (x^{\frac{1}{3}})^{10} (y^{\frac{1}{3}})^{2} + \frac{12 !}{3 ! ( 12 - 3 ) !} (x^{\frac{1}{3}})^{9} (y^{\frac{1}{3}})^{3} + \frac{12 !}{4 ! ( 12 - 4 ) !} (x^{\frac{1}{3}})^{8} (y^{\frac{1}{3}})^{4} + \frac{12 !}{5 ! ( 12 - 5 ) !} (x^{\frac{1}{3}})^{7} (y^{\frac{1}{3}})^{5} + \frac{12 !}{6 ! ( 12 - 6 ) !} (x^{\frac{1}{3}})^{6} (y^{\frac{1}{3}})^{6} + \frac{12 !}{7 ! ( 12 - 7 ) !} (x^{\frac{1}{3}})^{5} (y^{\frac{1}{3}})^{7} + \frac{12 !}{8 ! ( 12 - 8 ) !} (x^{\frac{1}{3}})^{4} (y^{\frac{1}{3}})^{8} + \frac{12 !}{9 ! ( 12 - 9 ) !} (x^{\frac{1}{3}})^{3} (y^{\frac{1}{3}})^{9} + \frac{12 !}{10 ! ( 12 - 10 ) !} (x^{\frac{1}{3}})^{2} (y^{\frac{1}{3}})^{10} + \frac{12 !}{11 ! ( 12 - 11 ) !} (x^{\frac{1}{3}})^{1} (y^{\frac{1}{3}})^{11} + \frac{12 !}{12 ! ( 12 - 12 ) !} (x^{\frac{1}{3}})^{0} (y^{\frac{1}{3}})^{12}

簡素化

\frac{12 !}{0 ! ( 12 - 0 ) !} (x^{\frac{1}{3}})^{12} (y^{\frac{1}{3}})^{0} : x^{4}

簡素化

\frac{12 !}{1 ! ( 12 - 1 ) !} (x^{\frac{1}{3}})^{11} (y^{\frac{1}{3}})^{1} : 12 x^{\frac{11}{3}} y^{\frac{1}{3}}

簡素化

\frac{12 !}{2 ! ( 12 - 2 ) !} (x^{\frac{1}{3}})^{10} (y^{\frac{1}{3}})^{2} : 66 x^{\frac{10}{3}} y^{\frac{2}{3}}

簡素化

\frac{12 !}{3 ! ( 12 - 3 ) !} (x^{\frac{1}{3}})^{9} (y^{\frac{1}{3}})^{3} : 220 x^{3} y

簡素化

\frac{12 !}{4 ! ( 12 - 4 ) !} (x^{\frac{1}{3}})^{8} (y^{\frac{1}{3}})^{4} : 495 x^{\frac{8}{3}} y^{\frac{4}{3}}

簡素化

\frac{12 !}{5 ! ( 12 - 5 ) !} (x^{\frac{1}{3}})^{7} (y^{\frac{1}{3}})^{5} : 792 x^{\frac{7}{3}} y^{\frac{5}{3}}

簡素化

\frac{12 !}{6 ! ( 12 - 6 ) !} (x^{\frac{1}{3}})^{6} (y^{\frac{1}{3}})^{6} : 924 x^{2} y^{2}

簡素化

\frac{12 !}{7 ! ( 12 - 7 ) !} (x^{\frac{1}{3}})^{5} (y^{\frac{1}{3}})^{7} : 792 x^{\frac{5}{3}} y^{\frac{7}{3}}

簡素化

\frac{12 !}{8 ! ( 12 - 8 ) !} (x^{\frac{1}{3}})^{4} (y^{\frac{1}{3}})^{8} : 495 x^{\frac{4}{3}} y^{\frac{8}{3}}

簡素化

\frac{12 !}{9 ! ( 12 - 9 ) !} (x^{\frac{1}{3}})^{3} (y^{\frac{1}{3}})^{9} : 220 x y^{3}

簡素化

\frac{12 !}{10 ! ( 12 - 10 ) !} (x^{\frac{1}{3}})^{2} (y^{\frac{1}{3}})^{10} : 66 x^{\frac{2}{3}} y^{\frac{10}{3}}

簡素化

\frac{12 !}{11 ! ( 12 - 11 ) !} (x^{\frac{1}{3}})^{1} (y^{\frac{1}{3}})^{11} : 12 x^{\frac{1}{3}} y^{\frac{11}{3}}

簡素化

\frac{12 !}{12 ! ( 12 - 12 ) !} (x^{\frac{1}{3}})^{0} (y^{\frac{1}{3}})^{12} : y^{4}

= x^{4} + 12 x^{\frac{11}{3}} y^{\frac{1}{3}} + 66 x^{\frac{10}{3}} y^{\frac{2}{3}} + 220 x^{3} y + 495 x^{\frac{8}{3}} y^{\frac{4}{3}} + 792 x^{\frac{7}{3}} y^{\frac{5}{3}} + 924 x^{2} y^{2} + 792 x^{\frac{5}{3}} y^{\frac{7}{3}} + 495 x^{\frac{4}{3}} y^{\frac{8}{3}} + 220 x y^{3} + 66 x^{\frac{2}{3}} y^{\frac{10}{3}} + 12 x^{\frac{1}{3}} y^{\frac{11}{3}} + y^{4}

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