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Beliebt Algebra >

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

Lösung

erweitern

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

Lösung

729 x^{12} + 486 x^{11} + 135 x^{10} + 20 x^{9} + \frac{5 x ^{8}}{3} + \frac{2 x ^{7}}{27} + \frac{x ^{6}}{729}

Schritte zur Lösung

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

Wende Binomialsetz an:

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

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

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

Erweitere Summation

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

Vereinfache

\frac{6 !}{0 ! ( 6 - 0 ) !} (3 x^{2})^{6} (\frac{x}{3})^{0} : 729 x^{12}

Vereinfache

\frac{6 !}{1 ! ( 6 - 1 ) !} (3 x^{2})^{5} (\frac{x}{3})^{1} : 486 x^{11}

Vereinfache

\frac{6 !}{2 ! ( 6 - 2 ) !} (3 x^{2})^{4} (\frac{x}{3})^{2} : 135 x^{10}

Vereinfache

\frac{6 !}{3 ! ( 6 - 3 ) !} (3 x^{2})^{3} (\frac{x}{3})^{3} : 20 x^{9}

Vereinfache

\frac{6 !}{4 ! ( 6 - 4 ) !} (3 x^{2})^{2} (\frac{x}{3})^{4} : \frac{5 x ^{8}}{3}

Vereinfache

\frac{6 !}{5 ! ( 6 - 5 ) !} (3 x^{2})^{1} (\frac{x}{3})^{5} : \frac{2 x ^{7}}{27}

Vereinfache

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

= 729 x^{12} + 486 x^{11} + 135 x^{10} + 20 x^{9} + \frac{5 x ^{8}}{3} + \frac{2 x ^{7}}{27} + \frac{x ^{6}}{729}

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