erweitern (x+2/x)^4

Lösung

erweitern

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

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

Schritte zur Lösung

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

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

Erweitere Summation

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

Vereinfache

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

Vereinfache

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

Vereinfache

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

Vereinfache

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

Vereinfache

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

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

e x p an d y^{'} (x) - (x - 1) y^{2} + (1 - 2 x) y + x

e x p an d \frac{x ^{2} + x - 25}{( x + 1 ) ^{2} ( x ^{2} + 4 )}

e x p an d lo g_{10} (8) x^{3}

e x p an d 4 (sin (\frac{19 π}{90}) + cos (\frac{17 π}{45})) e c^{8}

e x p an d \frac{2}{y} + \frac{1}{y - 5}