확장 (4/16 x+5/16 y+3/16 z+4/16 w)^4

해법

확장

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

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

솔루션 단계

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

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

간소화하다 :

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

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

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

확장 :

\frac{16 x ^{2} + 16 w ^{2} + 25 y ^{2} + 9 z ^{2}}{256} + \frac{5 x y + 3 x z + 5 y w + 3 z w}{32} + \frac{x w}{8} + \frac{15 yz}{128}

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

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

확장 :

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

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

괄호 배포

간소화하다 :

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

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