erweitern (x^2-2/x)^4

Lösung

erweitern

(x^{2} - \frac{2}{x})^{4}

x^{8} - 8 x^{5} + 24 x^{2} - \frac{32}{x} + \frac{16}{x ^{4}}

Schritte zur Lösung

(x^{2} - \frac{2}{x})^{4}

Wende Binomialsetz an:

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

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

= i = 0 \sum 4 (i 4) (x^{2})^{(4 - i)} (- \frac{2}{x})^{i}

Erweitere Summation

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

Vereinfache

\frac{4 !}{0 ! ( 4 - 0 ) !} (x^{2})^{4} (- \frac{2}{x})^{0} : x^{8}

Vereinfache

\frac{4 !}{1 ! ( 4 - 1 ) !} (x^{2})^{3} (- \frac{2}{x})^{1} : - 8 x^{5}

Vereinfache

\frac{4 !}{2 ! ( 4 - 2 ) !} (x^{2})^{2} (- \frac{2}{x})^{2} : 24 x^{2}

Vereinfache

\frac{4 !}{3 ! ( 4 - 3 ) !} (x^{2})^{1} (- \frac{2}{x})^{3} : - \frac{32}{x}

Vereinfache

\frac{4 !}{4 ! ( 4 - 4 ) !} (x^{2})^{0} (- \frac{2}{x})^{4} : \frac{16}{x ^{4}}

= x^{8} - 8 x^{5} + 24 x^{2} - \frac{32}{x} + \frac{16}{x ^{4}}

e x p an d (2 A x + B) e^{x} + e^{x} (A x^{2} + B x)

e x p an d \frac{( 6 - x ) ^{2}}{9}

e x p an d a (arctan (wT) - arctan (w Y))

e x p an d a^{2} b^{2} - c^{2} d^{2} (sin (2 α) + 1)

e x p an d (1 - L) \cdot (1 - t) \cdot w