展開する (4/16 x+5/16 y+3/16 z+4/16 u)^4

解

展開する

(\frac{4}{16} x + \frac{5}{16} y + \frac{3}{16} z + \frac{4}{16} u)^{4}

\frac{16 x ^{4} + 16 u ^{4} + 16 x ^{2} u ^{2} + 25 x ^{2} y ^{2} + 9 x ^{2} z ^{2} + 32 x ^{3} u + 32 x u ^{3} + 16 u ^{2} x ^{2} + 25 u ^{2} y ^{2} + 9 u ^{2} z ^{2} + 40 x ^{3} y + 24 x ^{3} z + 40 y u ^{3} + 24 z u ^{3} + 125 x y ^{3} + 125 y ^{3} u + 27 x z ^{3} + 27 z ^{3} u + 50 x y ^{2} u + 18 x z ^{2} u + 40 x ^{2} y u + 24 x ^{2} z u + 40 x y u ^{2} + 24 x z u ^{2} + 225 x y ^{2} z + 225 y ^{2} z u + 135 x y z ^{2} + 135 y z ^{2} u}{4096} + \frac{625 y ^{4} + 400 y ^{2} x ^{2} + 400 y ^{2} u ^{2} + 225 y ^{2} z ^{2} + 750 y ^{3} z + 270 y z ^{3} + 81 z ^{4} + 144 z ^{2} x ^{2} + 144 z ^{2} u ^{2} + 225 z ^{2} y ^{2} + 480 x ^{2} yz + 480 yz u ^{2}}{65536} + \frac{5 x ^{3} y + 3 x ^{3} z + 5 y u ^{3} + 3 z u ^{3} + 25 x y u ^{2} + 15 x z u ^{2} + 25 x ^{2} y u + 15 x ^{2} z u}{512} + \frac{25 y ^{2} x ^{2} + 25 y ^{2} u ^{2} + 9 z ^{2} x ^{2} + 9 z ^{2} u ^{2} + 25 x ^{2} y ^{2} + 9 x ^{2} z ^{2} + 25 u ^{2} y ^{2} + 9 u ^{2} z ^{2} + 75 x ^{2} yz + 75 yz u ^{2} + 125 x y ^{2} u + 45 x z ^{2} u + 180 x yz u}{2048} + \frac{( x ^{2} u + x u ^{2} ) x + ( x ^{2} u + x u ^{2} ) u}{128} + \frac{5 ( 75 y ^{2} z + 45 y z ^{2} ) y + 3 ( 75 y ^{2} z + 45 y z ^{2} ) z}{32768}

解答ステップ

(\frac{4}{16} x + \frac{5}{16} y + \frac{3}{16} z + \frac{4}{16} u)^{4}

簡素化

(\frac{4}{16} x + \frac{5}{16} y + \frac{3}{16} z + \frac{4}{16} u)^{4} : (\frac{4}{16} x + \frac{5}{16} y + \frac{3}{16} z + \frac{4}{16} u) (\frac{4}{16} x + \frac{5}{16} y + \frac{3}{16} z + \frac{4}{16} u) (\frac{4}{16} x + \frac{5}{16} y + \frac{3}{16} z + \frac{4}{16} u) (\frac{4}{16} x + \frac{5}{16} y + \frac{3}{16} z + \frac{4}{16} u)

= (\frac{4}{16} x + \frac{5}{16} y + \frac{3}{16} z + \frac{4}{16} u) (\frac{4}{16} x + \frac{5}{16} y + \frac{3}{16} z + \frac{4}{16} u) (\frac{4}{16} x + \frac{5}{16} y + \frac{3}{16} z + \frac{4}{16} u) (\frac{4}{16} x + \frac{5}{16} y + \frac{3}{16} z + \frac{4}{16} u)

拡張

(\frac{4}{16} x + \frac{5}{16} y + \frac{3}{16} z + \frac{4}{16} u) (\frac{4}{16} x + \frac{5}{16} y + \frac{3}{16} z + \frac{4}{16} u) : \frac{16 x ^{2} + 16 u ^{2} + 25 y ^{2} + 9 z ^{2}}{256} + \frac{5 x y + 3 x z + 5 y u + 3 z u}{32} + \frac{xu}{8} + \frac{15 yz}{128}

= (\frac{16 x ^{2} + 16 u ^{2} + 25 y ^{2} + 9 z ^{2}}{256} + \frac{5 x y + 3 x z + 5 y u + 3 z u}{32} + \frac{xu}{8} + \frac{15 yz}{128}) (\frac{4}{16} x + \frac{5}{16} y + \frac{3}{16} z + \frac{4}{16} u) (\frac{4}{16} x + \frac{5}{16} y + \frac{3}{16} z + \frac{4}{16} u)

拡張

(\frac{16 x ^{2} + 16 u ^{2} + 25 y ^{2} + 9 z ^{2}}{256} + \frac{5 x y + 3 x z + 5 y u + 3 z u}{32} + \frac{xu}{8} + \frac{15 yz}{128}) (\frac{4}{16} x + \frac{5}{16} y + \frac{3}{16} z + \frac{4}{16} u) : \frac{( 16 x ^{2} + 16 u ^{2} + 25 y ^{2} + 9 z ^{2} ) x + ( 16 x ^{2} + 16 u ^{2} + 25 y ^{2} + 9 z ^{2} ) u}{1024} + \frac{5 ( 16 x ^{2} + 16 u ^{2} + 25 y ^{2} + 9 z ^{2} ) y + 3 ( 16 x ^{2} + 16 u ^{2} + 25 y ^{2} + 9 z ^{2} ) z}{4096} + \frac{5 x ^{2} y + 3 x ^{2} z + 5 y u ^{2} + 3 z u ^{2} + 15 x y u + 9 x z u}{128} + \frac{25 x y ^{2} + 25 y ^{2} u + 9 x z ^{2} + 9 z ^{2} u + 45 x yz + 45 yz u}{512} + \frac{x ^{2} u + x u ^{2}}{32} + \frac{75 y ^{2} z + 45 y z ^{2}}{2048}

= (\frac{( 16 x ^{2} + 16 u ^{2} + 25 y ^{2} + 9 z ^{2} ) x + ( 16 x ^{2} + 16 u ^{2} + 25 y ^{2} + 9 z ^{2} ) u}{1024} + \frac{5 ( 16 x ^{2} + 16 u ^{2} + 25 y ^{2} + 9 z ^{2} ) y + 3 ( 16 x ^{2} + 16 u ^{2} + 25 y ^{2} + 9 z ^{2} ) z}{4096} + \frac{5 x ^{2} y + 3 x ^{2} z + 5 y u ^{2} + 3 z u ^{2} + 15 x y u + 9 x z u}{128} + \frac{25 x y ^{2} + 25 y ^{2} u + 9 x z ^{2} + 9 z ^{2} u + 45 x yz + 45 yz u}{512} + \frac{x ^{2} u + x u ^{2}}{32} + \frac{75 y ^{2} z + 45 y z ^{2}}{2048}) (\frac{4}{16} x + \frac{5}{16} y + \frac{3}{16} z + \frac{4}{16} u)

括弧を分配する

簡素化

= \frac{16 x ^{4} + 16 u ^{4} + 16 x ^{2} u ^{2} + 25 x ^{2} y ^{2} + 9 x ^{2} z ^{2} + 32 x ^{3} u + 32 x u ^{3} + 16 u ^{2} x ^{2} + 25 u ^{2} y ^{2} + 9 u ^{2} z ^{2} + 40 x ^{3} y + 24 x ^{3} z + 40 y u ^{3} + 24 z u ^{3} + 125 x y ^{3} + 125 y ^{3} u + 27 x z ^{3} + 27 z ^{3} u + 50 x y ^{2} u + 18 x z ^{2} u + 40 x ^{2} y u + 24 x ^{2} z u + 40 x y u ^{2} + 24 x z u ^{2} + 225 x y ^{2} z + 225 y ^{2} z u + 135 x y z ^{2} + 135 y z ^{2} u}{4096} + \frac{625 y ^{4} + 400 y ^{2} x ^{2} + 400 y ^{2} u ^{2} + 225 y ^{2} z ^{2} + 750 y ^{3} z + 270 y z ^{3} + 81 z ^{4} + 144 z ^{2} x ^{2} + 144 z ^{2} u ^{2} + 225 z ^{2} y ^{2} + 480 x ^{2} yz + 480 yz u ^{2}}{65536} + \frac{5 x ^{3} y + 3 x ^{3} z + 5 y u ^{3} + 3 z u ^{3} + 25 x y u ^{2} + 15 x z u ^{2} + 25 x ^{2} y u + 15 x ^{2} z u}{512} + \frac{25 y ^{2} x ^{2} + 25 y ^{2} u ^{2} + 9 z ^{2} x ^{2} + 9 z ^{2} u ^{2} + 25 x ^{2} y ^{2} + 9 x ^{2} z ^{2} + 25 u ^{2} y ^{2} + 9 u ^{2} z ^{2} + 75 x ^{2} yz + 75 yz u ^{2} + 125 x y ^{2} u + 45 x z ^{2} u + 180 x yz u}{2048} + \frac{( x ^{2} u + x u ^{2} ) x + ( x ^{2} u + x u ^{2} ) u}{128} + \frac{5 ( 75 y ^{2} z + 45 y z ^{2} ) y + 3 ( 75 y ^{2} z + 45 y z ^{2} ) z}{32768}