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Beliebt Algebra >

erweitern (x+1/(2sqrt(x)))^{14}

Lösung

erweitern

(x + \frac{1}{2 x})^{14}

Lösung

x^{14} + 7 x^{25} + \frac{91 x ^{11}}{4} + \frac{91 x ^{19}}{2} + \frac{1001 x ^{8}}{16} + \frac{1001 x ^{13}}{16} + \frac{3003 x ^{5}}{64} + \frac{429 x ^{7}}{16} + \frac{3003 x ^{2}}{256} + \frac{1001 x}{256} + \frac{1001}{1024 x} + \frac{91}{512 x ^{5}} + \frac{91}{4096 x ^{4}} + \frac{7}{4096 x ^{11}} + \frac{1}{16384 x ^{7}}

Schritte zur Lösung

(x + \frac{1}{2 x})^{14}

Wende Binomialsetz an:

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

a = x, b = \frac{1}{2 x}

= i = 0 \sum 14 (i 14) x^{(14 - i)} (\frac{1}{2 x})^{i}

Erweitere Summation

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

Vereinfache

\frac{14 !}{0 ! ( 14 - 0 ) !} x^{14} (\frac{1}{2 x})^{0} : x^{14}

Vereinfache

\frac{14 !}{1 ! ( 14 - 1 ) !} x^{13} (\frac{1}{2 x})^{1} : 7 x^{\frac{25}{2}}

Vereinfache

\frac{14 !}{2 ! ( 14 - 2 ) !} x^{12} (\frac{1}{2 x})^{2} : \frac{91 x ^{11}}{4}

Vereinfache

\frac{14 !}{3 ! ( 14 - 3 ) !} x^{11} (\frac{1}{2 x})^{3} : \frac{91 x ^{11}}{2 ( x ) ^{3}}

Vereinfache

\frac{14 !}{4 ! ( 14 - 4 ) !} x^{10} (\frac{1}{2 x})^{4} : \frac{1001 x ^{8}}{16}

Vereinfache

\frac{14 !}{5 ! ( 14 - 5 ) !} x^{9} (\frac{1}{2 x})^{5} : \frac{1001 x ^{9}}{16 ( x ) ^{5}}

Vereinfache

\frac{14 !}{6 ! ( 14 - 6 ) !} x^{8} (\frac{1}{2 x})^{6} : \frac{3003 x ^{5}}{64}

Vereinfache

\frac{14 !}{7 ! ( 14 - 7 ) !} x^{7} (\frac{1}{2 x})^{7} : \frac{429 x ^{7}}{16 ( x ) ^{7}}

Vereinfache

\frac{14 !}{8 ! ( 14 - 8 ) !} x^{6} (\frac{1}{2 x})^{8} : \frac{3003 x ^{2}}{256}

Vereinfache

\frac{14 !}{9 ! ( 14 - 9 ) !} x^{5} (\frac{1}{2 x})^{9} : \frac{1001 x ^{5}}{256 ( x ) ^{9}}

Vereinfache

\frac{14 !}{10 ! ( 14 - 10 ) !} x^{4} (\frac{1}{2 x})^{10} : \frac{1001}{1024 x}

Vereinfache

\frac{14 !}{11 ! ( 14 - 11 ) !} x^{3} (\frac{1}{2 x})^{11} : \frac{91 x ^{3}}{512 ( x ) ^{11}}

Vereinfache

\frac{14 !}{12 ! ( 14 - 12 ) !} x^{2} (\frac{1}{2 x})^{12} : \frac{91}{4096 x ^{4}}

Vereinfache

\frac{14 !}{13 ! ( 14 - 13 ) !} x^{1} (\frac{1}{2 x})^{13} : \frac{7 x}{4096 ( x ) ^{13}}

Vereinfache

\frac{14 !}{14 ! ( 14 - 14 ) !} x^{0} (\frac{1}{2 x})^{14} : \frac{1}{16384 x ^{7}}

= x^{14} + 7 x^{\frac{25}{2}} + \frac{91 x ^{11}}{4} + \frac{91 x ^{11}}{2 ( x ) ^{3}} + \frac{1001 x ^{8}}{16} + \frac{1001 x ^{9}}{16 ( x ) ^{5}} + \frac{3003 x ^{5}}{64} + \frac{429 x ^{7}}{16 ( x ) ^{7}} + \frac{3003 x ^{2}}{256} + \frac{1001 x ^{5}}{256 ( x ) ^{9}} + \frac{1001}{1024 x} + \frac{91 x ^{3}}{512 ( x ) ^{11}} + \frac{91}{4096 x ^{4}} + \frac{7 x}{4096 ( x ) ^{13}} + \frac{1}{16384 x ^{7}}

Vereinfache

x^{14} + 7 x^{\frac{25}{2}} + \frac{91 x ^{11}}{4} + \frac{91 x ^{11}}{2 ( x ) ^{3}} + \frac{1001 x ^{8}}{16} + \frac{1001 x ^{9}}{16 ( x ) ^{5}} + \frac{3003 x ^{5}}{64} + \frac{429 x ^{7}}{16 ( x ) ^{7}} + \frac{3003 x ^{2}}{256} + \frac{1001 x ^{5}}{256 ( x ) ^{9}} + \frac{1001}{1024 x} + \frac{91 x ^{3}}{512 ( x ) ^{11}} + \frac{91}{4096 x ^{4}} + \frac{7 x}{4096 ( x ) ^{13}} + \frac{1}{16384 x ^{7}} : x^{14} + 7 x^{\frac{25}{2}} + \frac{91 x ^{11}}{4} + \frac{91 x ^{\frac{19}{2}}}{2} + \frac{1001 x ^{8}}{16} + \frac{1001 x ^{\frac{13}{2}}}{16} + \frac{3003 x ^{5}}{64} + \frac{429 x ^{\frac{7}{2}}}{16} + \frac{3003 x ^{2}}{256} + \frac{1001 x}{256} + \frac{1001}{1024 x} + \frac{91}{512 x ^{\frac{5}{2}}} + \frac{91}{4096 x ^{4}} + \frac{7}{4096 x ^{\frac{11}{2}}} + \frac{1}{16384 x ^{7}}

= x^{14} + 7 x^{\frac{25}{2}} + \frac{91 x ^{11}}{4} + \frac{91 x ^{\frac{19}{2}}}{2} + \frac{1001 x ^{8}}{16} + \frac{1001 x ^{\frac{13}{2}}}{16} + \frac{3003 x ^{5}}{64} + \frac{429 x ^{\frac{7}{2}}}{16} + \frac{3003 x ^{2}}{256} + \frac{1001 x}{256} + \frac{1001}{1024 x} + \frac{91}{512 x ^{\frac{5}{2}}} + \frac{91}{4096 x ^{4}} + \frac{7}{4096 x ^{\frac{11}{2}}} + \frac{1}{16384 x ^{7}}

Vereinfache

x^{14} + 7 x^{\frac{25}{2}} + \frac{91 x ^{11}}{4} + \frac{91 x ^{\frac{19}{2}}}{2} + \frac{1001 x ^{8}}{16} + \frac{1001 x ^{\frac{13}{2}}}{16} + \frac{3003 x ^{5}}{64} + \frac{429 x ^{\frac{7}{2}}}{16} + \frac{3003 x ^{2}}{256} + \frac{1001 x}{256} + \frac{1001}{1024 x} + \frac{91}{512 x ^{\frac{5}{2}}} + \frac{91}{4096 x ^{4}} + \frac{7}{4096 x ^{\frac{11}{2}}} + \frac{1}{16384 x ^{7}} : x^{14} + 7 x^{25} + \frac{91 x ^{11}}{4} + \frac{91 x ^{19}}{2} + \frac{1001 x ^{8}}{16} + \frac{1001 x ^{13}}{16} + \frac{3003 x ^{5}}{64} + \frac{429 x ^{7}}{16} + \frac{3003 x ^{2}}{256} + \frac{1001 x}{256} + \frac{1001}{1024 x} + \frac{91}{512 x ^{5}} + \frac{91}{4096 x ^{4}} + \frac{7}{4096 x ^{11}} + \frac{1}{16384 x ^{7}}

= x^{14} + 7 x^{25} + \frac{91 x ^{11}}{4} + \frac{91 x ^{19}}{2} + \frac{1001 x ^{8}}{16} + \frac{1001 x ^{13}}{16} + \frac{3003 x ^{5}}{64} + \frac{429 x ^{7}}{16} + \frac{3003 x ^{2}}{256} + \frac{1001 x}{256} + \frac{1001}{1024 x} + \frac{91}{512 x ^{5}} + \frac{91}{4096 x ^{4}} + \frac{7}{4096 x ^{11}} + \frac{1}{16384 x ^{7}}

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