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Beliebt Algebra >

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

Lösung

erweitern

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

Lösung

\frac{16 x ^{8}}{y ^{4}} + \frac{32 x ^{5}}{y} + 24 x^{2} y^{2} + \frac{8 y ^{5}}{x} + \frac{y ^{8}}{x ^{4}}

Schritte zur Lösung

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

Wende Binomialsetz an:

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

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

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

Erweitere Summation

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

Vereinfache

\frac{4 !}{0 ! ( 4 - 0 ) !} (\frac{2 x ^{2}}{y})^{4} (\frac{y ^{2}}{x})^{0} : \frac{16 x ^{8}}{y ^{4}}

Vereinfache

\frac{4 !}{1 ! ( 4 - 1 ) !} (\frac{2 x ^{2}}{y})^{3} (\frac{y ^{2}}{x})^{1} : \frac{32 x ^{5}}{y}

Vereinfache

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

Vereinfache

\frac{4 !}{3 ! ( 4 - 3 ) !} (\frac{2 x ^{2}}{y})^{1} (\frac{y ^{2}}{x})^{3} : \frac{8 y ^{5}}{x}

Vereinfache

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

= \frac{16 x ^{8}}{y ^{4}} + \frac{32 x ^{5}}{y} + 24 x^{2} y^{2} + \frac{8 y ^{5}}{x} + \frac{y ^{8}}{x ^{4}}

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