erweitern (2x+(15)/(x^2))^4

Lösung

erweitern

(2 x + \frac{15}{x ^{2}})^{4}

16 x^{4} + 480 x + \frac{5400}{x ^{2}} + \frac{27000}{x ^{5}} + \frac{50625}{x ^{8}}

Schritte zur Lösung

(2 x + \frac{15}{x ^{2}})^{4}

Wende Binomialsetz an:

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

a = 2 x, b = \frac{15}{x ^{2}}

= i = 0 \sum 4 (i 4) (2 x)^{(4 - i)} (\frac{15}{x ^{2}})^{i}

Erweitere Summation

= \frac{4 !}{0 ! ( 4 - 0 ) !} (2 x)^{4} (\frac{15}{x ^{2}})^{0} + \frac{4 !}{1 ! ( 4 - 1 ) !} (2 x)^{3} (\frac{15}{x ^{2}})^{1} + \frac{4 !}{2 ! ( 4 - 2 ) !} (2 x)^{2} (\frac{15}{x ^{2}})^{2} + \frac{4 !}{3 ! ( 4 - 3 ) !} (2 x)^{1} (\frac{15}{x ^{2}})^{3} + \frac{4 !}{4 ! ( 4 - 4 ) !} (2 x)^{0} (\frac{15}{x ^{2}})^{4}

Vereinfache

\frac{4 !}{0 ! ( 4 - 0 ) !} (2 x)^{4} (\frac{15}{x ^{2}})^{0} : 16 x^{4}

Vereinfache

\frac{4 !}{1 ! ( 4 - 1 ) !} (2 x)^{3} (\frac{15}{x ^{2}})^{1} : 480 x

Vereinfache

\frac{4 !}{2 ! ( 4 - 2 ) !} (2 x)^{2} (\frac{15}{x ^{2}})^{2} : \frac{5400}{x ^{2}}

Vereinfache

\frac{4 !}{3 ! ( 4 - 3 ) !} (2 x)^{1} (\frac{15}{x ^{2}})^{3} : \frac{27000}{x ^{5}}

Vereinfache

\frac{4 !}{4 ! ( 4 - 4 ) !} (2 x)^{0} (\frac{15}{x ^{2}})^{4} : \frac{50625}{x ^{8}}

= 16 x^{4} + 480 x + \frac{5400}{x ^{2}} + \frac{27000}{x ^{5}} + \frac{50625}{x ^{8}}

e x p an d \frac{x ^{5} + 1}{x ( x - 1 ) ( x ^{2} + 1 ) ^{2}}

e x p an d 5 (x + y) - 6 x + y - 8

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

e x p an d (1 - sin (x)) (1 + cos (x))

e x p an d 3 \times 9