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Beliebt Algebra >

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

Lösung

erweitern

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

Lösung

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

Vereinfache

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

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

Schreibe

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

um:

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

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

Schreibe

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

um:

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

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

Setze Klammern

Vereinfache

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

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