erweitern (4/16 x+5/16 y+3/16 z+4/16 t)^4

Lösung

erweitern

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

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

Schritte zur Lösung

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

Vereinfache

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

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

Schreibe

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

um:

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

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

Schreibe

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

um:

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

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

Setze Klammern

Vereinfache

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

s im pl i f y (4 + 2 i)^{5}

s im pl i f y e^{(1 + 2 i) t}

e x p an d \frac{π h ^{2} ( 3 r - h )}{3}

e x p an d - 2.5 (25 x^{2} + 361)^{\frac{1}{2}} + 25 (x + 2.5)

e x p an d - \frac{1}{( x + 4 ) ^{2}}