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Beliebt Algebra >

erweitern (x-2/(sqrt(x)))^{15}

Lösung

erweitern

(x - \frac{2}{x})^{15}

Lösung

x^{15} - 30 x^{27} + 420 x^{12} - 3640 x^{21} - 96096 x^{15} - 823680 x^{9} - 2562560 x^{3} + 21840 x^{9} + 320320 x^{6} + 1647360 x^{3} + 3075072 - \frac{2795520}{x ^{3}} + \frac{1863680}{x ^{3}} - \frac{860160}{x ^{9}} + \frac{245760}{x ^{6}} - \frac{32768}{x ^{15}}

Schritte zur Lösung

(x - \frac{2}{x})^{15}

Wende Binomialsetz an:

(a + b)^{n} = i = 0 \sum n (i n) a^{(n - i)} b^{i}

a = x, b = - \frac{2}{x}

= i = 0 \sum 15 (i 15) x^{(15 - i)} (- \frac{2}{x})^{i}

Erweitere Summation

= \frac{15 !}{0 ! ( 15 - 0 ) !} x^{15} (- \frac{2}{x})^{0} + \frac{15 !}{1 ! ( 15 - 1 ) !} x^{14} (- \frac{2}{x})^{1} + \frac{15 !}{2 ! ( 15 - 2 ) !} x^{13} (- \frac{2}{x})^{2} + \frac{15 !}{3 ! ( 15 - 3 ) !} x^{12} (- \frac{2}{x})^{3} + \frac{15 !}{4 ! ( 15 - 4 ) !} x^{11} (- \frac{2}{x})^{4} + \frac{15 !}{5 ! ( 15 - 5 ) !} x^{10} (- \frac{2}{x})^{5} + \frac{15 !}{6 ! ( 15 - 6 ) !} x^{9} (- \frac{2}{x})^{6} + \frac{15 !}{7 ! ( 15 - 7 ) !} x^{8} (- \frac{2}{x})^{7} + \frac{15 !}{8 ! ( 15 - 8 ) !} x^{7} (- \frac{2}{x})^{8} + \frac{15 !}{9 ! ( 15 - 9 ) !} x^{6} (- \frac{2}{x})^{9} + \frac{15 !}{10 ! ( 15 - 10 ) !} x^{5} (- \frac{2}{x})^{10} + \frac{15 !}{11 ! ( 15 - 11 ) !} x^{4} (- \frac{2}{x})^{11} + \frac{15 !}{12 ! ( 15 - 12 ) !} x^{3} (- \frac{2}{x})^{12} + \frac{15 !}{13 ! ( 15 - 13 ) !} x^{2} (- \frac{2}{x})^{13} + \frac{15 !}{14 ! ( 15 - 14 ) !} x^{1} (- \frac{2}{x})^{14} + \frac{15 !}{15 ! ( 15 - 15 ) !} x^{0} (- \frac{2}{x})^{15}

Vereinfache

\frac{15 !}{0 ! ( 15 - 0 ) !} x^{15} (- \frac{2}{x})^{0} : x^{15}

Vereinfache

\frac{15 !}{1 ! ( 15 - 1 ) !} x^{14} (- \frac{2}{x})^{1} : - 30 x^{\frac{27}{2}}

Vereinfache

\frac{15 !}{2 ! ( 15 - 2 ) !} x^{13} (- \frac{2}{x})^{2} : 420 x^{12}

Vereinfache

\frac{15 !}{3 ! ( 15 - 3 ) !} x^{12} (- \frac{2}{x})^{3} : - \frac{3640 x ^{12}}{( x ) ^{3}}

Vereinfache

\frac{15 !}{4 ! ( 15 - 4 ) !} x^{11} (- \frac{2}{x})^{4} : 21840 x^{9}

Vereinfache

\frac{15 !}{5 ! ( 15 - 5 ) !} x^{10} (- \frac{2}{x})^{5} : - \frac{96096 x ^{10}}{( x ) ^{5}}

Vereinfache

\frac{15 !}{6 ! ( 15 - 6 ) !} x^{9} (- \frac{2}{x})^{6} : 320320 x^{6}

Vereinfache

\frac{15 !}{7 ! ( 15 - 7 ) !} x^{8} (- \frac{2}{x})^{7} : - \frac{823680 x ^{8}}{( x ) ^{7}}

Vereinfache

\frac{15 !}{8 ! ( 15 - 8 ) !} x^{7} (- \frac{2}{x})^{8} : 1647360 x^{3}

Vereinfache

\frac{15 !}{9 ! ( 15 - 9 ) !} x^{6} (- \frac{2}{x})^{9} : - \frac{2562560 x ^{6}}{( x ) ^{9}}

Vereinfache

\frac{15 !}{10 ! ( 15 - 10 ) !} x^{5} (- \frac{2}{x})^{10} : 3075072

Vereinfache

\frac{15 !}{11 ! ( 15 - 11 ) !} x^{4} (- \frac{2}{x})^{11} : - \frac{2795520 x ^{4}}{( x ) ^{11}}

Vereinfache

\frac{15 !}{12 ! ( 15 - 12 ) !} x^{3} (- \frac{2}{x})^{12} : \frac{1863680}{x ^{3}}

Vereinfache

\frac{15 !}{13 ! ( 15 - 13 ) !} x^{2} (- \frac{2}{x})^{13} : - \frac{860160 x ^{2}}{( x ) ^{13}}

Vereinfache

\frac{15 !}{14 ! ( 15 - 14 ) !} x^{1} (- \frac{2}{x})^{14} : \frac{245760}{x ^{6}}

Vereinfache

\frac{15 !}{15 ! ( 15 - 15 ) !} x^{0} (- \frac{2}{x})^{15} : - \frac{32768}{( x ) ^{15}}

= x^{15} - 30 x^{\frac{27}{2}} + 420 x^{12} - \frac{3640 x ^{12}}{( x ) ^{3}} + 21840 x^{9} - \frac{96096 x ^{10}}{( x ) ^{5}} + 320320 x^{6} - \frac{823680 x ^{8}}{( x ) ^{7}} + 1647360 x^{3} - \frac{2562560 x ^{6}}{( x ) ^{9}} + 3075072 - \frac{2795520 x ^{4}}{( x ) ^{11}} + \frac{1863680}{x ^{3}} - \frac{860160 x ^{2}}{( x ) ^{13}} + \frac{245760}{x ^{6}} - \frac{32768}{( x ) ^{15}}

\frac{3640 x ^{12}}{( x ) ^{3}} = \frac{3640 x ^{11}}{x}

\frac{96096 x ^{10}}{( x ) ^{5}} = \frac{96096 x ^{8}}{x}

\frac{823680 x ^{8}}{( x ) ^{7}} = \frac{823680 x ^{5}}{x}

\frac{2562560 x ^{6}}{( x ) ^{9}} = \frac{2562560 x ^{2}}{x}

\frac{2795520 x ^{4}}{( x ) ^{11}} = \frac{2795520}{x ^{\frac{3}{2}}}

\frac{860160 x ^{2}}{( x ) ^{13}} = \frac{860160}{x ^{\frac{9}{2}}}

\frac{32768}{( x ) ^{15}} = \frac{32768}{x ^{\frac{15}{2}}}

= x^{15} - 30 x^{\frac{27}{2}} + 420 x^{12} - \frac{3640 x ^{11}}{x} + 21840 x^{9} - \frac{96096 x ^{8}}{x} + 320320 x^{6} - \frac{823680 x ^{5}}{x} + 1647360 x^{3} - \frac{2562560 x ^{2}}{x} + 3075072 - \frac{2795520}{x ^{\frac{3}{2}}} + \frac{1863680}{x ^{3}} - \frac{860160}{x ^{\frac{9}{2}}} + \frac{245760}{x ^{6}} - \frac{32768}{x ^{\frac{15}{2}}}

Füge zusammen:

x^{15} - 30 x^{\frac{27}{2}} + 420 x^{12} - \frac{3640 x ^{11}}{x} + 21840 x^{9} - \frac{96096 x ^{8}}{x} + 320320 x^{6} - \frac{823680 x ^{5}}{x} + 1647360 x^{3} - \frac{2562560 x ^{2}}{x} + 3075072 - \frac{2795520}{x ^{\frac{3}{2}}} + \frac{1863680}{x ^{3}} - \frac{860160}{x ^{\frac{9}{2}}} + \frac{245760}{x ^{6}} - \frac{32768}{x ^{\frac{15}{2}}} : x^{15} - 30 x^{\frac{27}{2}} + 420 x^{12} - 104 x^{\frac{3}{2}} (35 x^{9} + 924 x^{6} + 7920 x^{3} + 24640) + 21840 x^{9} + 320320 x^{6} + 1647360 x^{3} + 3075072 - \frac{2795520}{x ^{\frac{3}{2}}} + \frac{1863680}{x ^{3}} - \frac{860160}{x ^{\frac{9}{2}}} + \frac{245760}{x ^{6}} - \frac{32768}{x ^{\frac{15}{2}}}

= x^{15} - 30 x^{\frac{27}{2}} + 420 x^{12} - 104 x^{\frac{3}{2}} (35 x^{9} + 924 x^{6} + 7920 x^{3} + 24640) + 21840 x^{9} + 320320 x^{6} + 1647360 x^{3} + 3075072 - \frac{2795520}{x ^{\frac{3}{2}}} + \frac{1863680}{x ^{3}} - \frac{860160}{x ^{\frac{9}{2}}} + \frac{245760}{x ^{6}} - \frac{32768}{x ^{\frac{15}{2}}}

Schreibe

- 104 x^{\frac{3}{2}} (35 x^{9} + 924 x^{6} + 7920 x^{3} + 24640)

um:

- 3640 x^{\frac{21}{2}} - 96096 x^{\frac{15}{2}} - 823680 x^{\frac{9}{2}} - 2562560 x^{\frac{3}{2}}

= x^{15} - 30 x^{\frac{27}{2}} + 420 x^{12} - 3640 x^{\frac{21}{2}} - 96096 x^{\frac{15}{2}} - 823680 x^{\frac{9}{2}} - 2562560 x^{\frac{3}{2}} + 21840 x^{9} + 320320 x^{6} + 1647360 x^{3} + 3075072 - \frac{2795520}{x ^{\frac{3}{2}}} + \frac{1863680}{x ^{3}} - \frac{860160}{x ^{\frac{9}{2}}} + \frac{245760}{x ^{6}} - \frac{32768}{x ^{\frac{15}{2}}}

Vereinfache

x^{15} - 30 x^{\frac{27}{2}} + 420 x^{12} - 3640 x^{\frac{21}{2}} - 96096 x^{\frac{15}{2}} - 823680 x^{\frac{9}{2}} - 2562560 x^{\frac{3}{2}} + 21840 x^{9} + 320320 x^{6} + 1647360 x^{3} + 3075072 - \frac{2795520}{x ^{\frac{3}{2}}} + \frac{1863680}{x ^{3}} - \frac{860160}{x ^{\frac{9}{2}}} + \frac{245760}{x ^{6}} - \frac{32768}{x ^{\frac{15}{2}}} : x^{15} - 30 x^{27} + 420 x^{12} - 3640 x^{21} - 96096 x^{15} - 823680 x^{9} - 2562560 x^{3} + 21840 x^{9} + 320320 x^{6} + 1647360 x^{3} + 3075072 - \frac{2795520}{x ^{3}} + \frac{1863680}{x ^{3}} - \frac{860160}{x ^{9}} + \frac{245760}{x ^{6}} - \frac{32768}{x ^{15}}

= x^{15} - 30 x^{27} + 420 x^{12} - 3640 x^{21} - 96096 x^{15} - 823680 x^{9} - 2562560 x^{3} + 21840 x^{9} + 320320 x^{6} + 1647360 x^{3} + 3075072 - \frac{2795520}{x ^{3}} + \frac{1863680}{x ^{3}} - \frac{860160}{x ^{9}} + \frac{245760}{x ^{6}} - \frac{32768}{x ^{15}}

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