erweitern (3x-2/(x^2))^6

Lösung

erweitern

(3 x - \frac{2}{x ^{2}})^{6}

729 x^{6} - 2916 x^{3} + 4860 - \frac{4320}{x ^{3}} + \frac{2160}{x ^{6}} - \frac{576}{x ^{9}} + \frac{64}{x ^{12}}

Schritte zur Lösung

(3 x - \frac{2}{x ^{2}})^{6}

Wende Binomialsetz an:

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

a = 3 x, b = - \frac{2}{x ^{2}}

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

Erweitere Summation

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

Vereinfache

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

Vereinfache

\frac{6 !}{1 ! ( 6 - 1 ) !} (3 x)^{5} (- \frac{2}{x ^{2}})^{1} : - 2916 x^{3}

Vereinfache

\frac{6 !}{2 ! ( 6 - 2 ) !} (3 x)^{4} (- \frac{2}{x ^{2}})^{2} : 4860

Vereinfache

\frac{6 !}{3 ! ( 6 - 3 ) !} (3 x)^{3} (- \frac{2}{x ^{2}})^{3} : - \frac{4320}{x ^{3}}

Vereinfache

\frac{6 !}{4 ! ( 6 - 4 ) !} (3 x)^{2} (- \frac{2}{x ^{2}})^{4} : \frac{2160}{x ^{6}}

Vereinfache

\frac{6 !}{5 ! ( 6 - 5 ) !} (3 x)^{1} (- \frac{2}{x ^{2}})^{5} : - \frac{576}{x ^{9}}

Vereinfache

\frac{6 !}{6 ! ( 6 - 6 ) !} (3 x)^{0} (- \frac{2}{x ^{2}})^{6} : \frac{64}{x ^{12}}

= 729 x^{6} - 2916 x^{3} + 4860 - \frac{4320}{x ^{3}} + \frac{2160}{x ^{6}} - \frac{576}{x ^{9}} + \frac{64}{x ^{12}}

e x p an d \frac{( π \cdot ( 60 - h ) \cdot h ^{2} )}{3}

e x p an d (\frac{x y ^{2}}{2} = \frac{y}{x} + C) \cdot 2 x

\frac{d}{d x} ((z) (y z + x ln (y) - z^{2}))

e x p an d \frac{20 ( s + 5 )}{s + 20}

e x p an d \frac{( x ^{4} + 6 x - 1 )}{( x + 1 ) ( x ^{2} - 2 x + 4 )}