erweitern (y+1/(y^2))^6

Lösung

erweitern

(y + \frac{1}{y ^{2}})^{6}

y^{6} + 6 y^{3} + 15 + \frac{20}{y ^{3}} + \frac{15}{y ^{6}} + \frac{6}{y ^{9}} + \frac{1}{y ^{12}}

Schritte zur Lösung

(y + \frac{1}{y ^{2}})^{6}

Wende Binomialsetz an:

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

a = y, b = \frac{1}{y ^{2}}

= i = 0 \sum 6 (i 6) y^{(6 - i)} (\frac{1}{y ^{2}})^{i}

Erweitere Summation

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

Vereinfache

\frac{6 !}{0 ! ( 6 - 0 ) !} y^{6} (\frac{1}{y ^{2}})^{0} : y^{6}

Vereinfache

\frac{6 !}{1 ! ( 6 - 1 ) !} y^{5} (\frac{1}{y ^{2}})^{1} : 6 y^{3}

Vereinfache

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

Vereinfache

\frac{6 !}{3 ! ( 6 - 3 ) !} y^{3} (\frac{1}{y ^{2}})^{3} : \frac{20}{y ^{3}}

Vereinfache

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

Vereinfache

\frac{6 !}{5 ! ( 6 - 5 ) !} y^{1} (\frac{1}{y ^{2}})^{5} : \frac{6}{y ^{9}}

Vereinfache

\frac{6 !}{6 ! ( 6 - 6 ) !} y^{0} (\frac{1}{y ^{2}})^{6} : \frac{1}{y ^{12}}

= y^{6} + 6 y^{3} + 15 + \frac{20}{y ^{3}} + \frac{15}{y ^{6}} + \frac{6}{y ^{9}} + \frac{1}{y ^{12}}

s im pl i f y (x - 4) \times (x + 2)

e x p an d \frac{\frac{d}{d x} ( ( x + 3 ) ( x - 1 ) )}{( x + 2 )}

e x p an d \frac{( 4 - 2 x ) ^{2} ( 1 + 2 x ) ^{2}}{( 1 - 3 x ) ^{3} ( 1 - x )}

e x p an d (1 + \frac{2 x}{y})^{3}

e x p an d 3 (t + Δ t)^{2} (- sin (Δ t) + cos (Δ t))