erweitern (1+sqrt(x))^6

Lösung

erweitern

(1 + x)^{6}

1 + 6 x + 15 x + 20 x x + 15 x^{2} + 6 x^{2} x + x^{3}

Schritte zur Lösung

(1 + x)^{6}

Wende Binomialsetz an:

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

a = 1, b = x

= i = 0 \sum 6 (i 6) \cdot 1^{(6 - i)} (x)^{i}

Erweitere Summation

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

Vereinfache

\frac{6 !}{0 ! ( 6 - 0 ) !} \cdot 1^{6} (x)^{0} : 1

Vereinfache

\frac{6 !}{1 ! ( 6 - 1 ) !} \cdot 1^{5} (x)^{1} : 6 x

Vereinfache

\frac{6 !}{2 ! ( 6 - 2 ) !} \cdot 1^{4} (x)^{2} : 15 x

Vereinfache

\frac{6 !}{3 ! ( 6 - 3 ) !} \cdot 1^{3} (x)^{3} : 20 (x)^{3}

Vereinfache

\frac{6 !}{4 ! ( 6 - 4 ) !} \cdot 1^{2} (x)^{4} : 15 x^{2}

Vereinfache

\frac{6 !}{5 ! ( 6 - 5 ) !} \cdot 1^{1} (x)^{5} : 6 (x)^{5}

Vereinfache

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

= 1 + 6 x + 15 x + 20 (x)^{3} + 15 x^{2} + 6 (x)^{5} + x^{3}

(x)^{3} = x x

(x)^{5} = x^{2} x

= 1 + 6 x + 15 x + 20 x x + 15 x^{2} + 6 x^{2} x + x^{3}

e x p an d \frac{2 ( s ^{2} + 4 )}{s ^{2} + 2}

e x p an d (x + 2) (x - 2) (x + \frac{17}{10}) (x - \frac{17}{10})

e x p an d x 2 - 1.695 x + 0.2583

e x p an d (9 + 3) (7 + 2)

e x p an d \frac{( 3 - w ) ^{2}}{2}