erweitern (3/(sqrt(2x))+1/y)^6

Lösung

erweitern

(\frac{3}{2 x} + \frac{1}{y})^{6}

\frac{729}{8 x ^{3}} + \frac{729}{2 2 y x ^{2} x} + \frac{1215}{4 x ^{2} y ^{2}} + \frac{135 2}{y ^{3} x x} + \frac{135}{2 x y ^{4}} + \frac{9 2}{y ^{5} x} + \frac{1}{y ^{6}}

Schritte zur Lösung

(\frac{3}{2 x} + \frac{1}{y})^{6}

Wende Binomialsetz an:

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

a = \frac{3}{2 x}, b = \frac{1}{y}

= i = 0 \sum 6 (i 6) (\frac{3}{2 x})^{(6 - i)} (\frac{1}{y})^{i}

Erweitere Summation

= \frac{6 !}{0 ! ( 6 - 0 ) !} (\frac{3}{2 x})^{6} (\frac{1}{y})^{0} + \frac{6 !}{1 ! ( 6 - 1 ) !} (\frac{3}{2 x})^{5} (\frac{1}{y})^{1} + \frac{6 !}{2 ! ( 6 - 2 ) !} (\frac{3}{2 x})^{4} (\frac{1}{y})^{2} + \frac{6 !}{3 ! ( 6 - 3 ) !} (\frac{3}{2 x})^{3} (\frac{1}{y})^{3} + \frac{6 !}{4 ! ( 6 - 4 ) !} (\frac{3}{2 x})^{2} (\frac{1}{y})^{4} + \frac{6 !}{5 ! ( 6 - 5 ) !} (\frac{3}{2 x})^{1} (\frac{1}{y})^{5} + \frac{6 !}{6 ! ( 6 - 6 ) !} (\frac{3}{2 x})^{0} (\frac{1}{y})^{6}

Vereinfache

\frac{6 !}{0 ! ( 6 - 0 ) !} (\frac{3}{2 x})^{6} (\frac{1}{y})^{0} : \frac{729}{8 x ^{3}}

Vereinfache

\frac{6 !}{1 ! ( 6 - 1 ) !} (\frac{3}{2 x})^{5} (\frac{1}{y})^{1} : \frac{729}{2 2 y ( x ) ^{5}}

Vereinfache

\frac{6 !}{2 ! ( 6 - 2 ) !} (\frac{3}{2 x})^{4} (\frac{1}{y})^{2} : \frac{1215}{4 x ^{2} y ^{2}}

Vereinfache

\frac{6 !}{3 ! ( 6 - 3 ) !} (\frac{3}{2 x})^{3} (\frac{1}{y})^{3} : \frac{135 2}{y ^{3} ( x ) ^{3}}

Vereinfache

\frac{6 !}{4 ! ( 6 - 4 ) !} (\frac{3}{2 x})^{2} (\frac{1}{y})^{4} : \frac{135}{2 x y ^{4}}

Vereinfache

\frac{6 !}{5 ! ( 6 - 5 ) !} (\frac{3}{2 x})^{1} (\frac{1}{y})^{5} : \frac{9 2}{y ^{5} x}

Vereinfache

\frac{6 !}{6 ! ( 6 - 6 ) !} (\frac{3}{2 x})^{0} (\frac{1}{y})^{6} : \frac{1}{y ^{6}}

= \frac{729}{8 x ^{3}} + \frac{729}{2 2 y ( x ) ^{5}} + \frac{1215}{4 x ^{2} y ^{2}} + \frac{135 2}{y ^{3} ( x ) ^{3}} + \frac{135}{2 x y ^{4}} + \frac{9 2}{y ^{5} x} + \frac{1}{y ^{6}}

Vereinfache

\frac{729}{8 x ^{3}} + \frac{729}{2 2 y ( x ) ^{5}} + \frac{1215}{4 x ^{2} y ^{2}} + \frac{135 2}{y ^{3} ( x ) ^{3}} + \frac{135}{2 x y ^{4}} + \frac{9 2}{y ^{5} x} + \frac{1}{y ^{6}} : \frac{729}{8 x ^{3}} + \frac{729}{2 2 y x ^{2} x} + \frac{1215}{4 x ^{2} y ^{2}} + \frac{135 2}{y ^{3} x x} + \frac{135}{2 x y ^{4}} + \frac{9 2}{y ^{5} x} + \frac{1}{y ^{6}}

= \frac{729}{8 x ^{3}} + \frac{729}{2 2 y x ^{2} x} + \frac{1215}{4 x ^{2} y ^{2}} + \frac{135 2}{y ^{3} x x} + \frac{135}{2 x y ^{4}} + \frac{9 2}{y ^{5} x} + \frac{1}{y ^{6}}

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