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Beliebt Algebra >

erweitern (1+2/3 x)^7(3+7x)^2

Lösung

erweitern

(1 + \frac{2}{3} x)^{7} (3 + 7 x)^{2}

Lösung

9 + 84 x + 329 x^{2} + 714 x^{3} + \frac{8596 x ^{4}}{9} + \frac{22232 x ^{5}}{27} + \frac{4144 x ^{6}}{9} + \frac{39328 x ^{7}}{243} + \frac{23744 x ^{8}}{729} + \frac{6272 x ^{9}}{2187}

Schritte zur Lösung

(1 + \frac{2}{3} x)^{7} (3 + 7 x)^{2}

Schreibe

(3 + 7 x)^{2}

um:

9 + 42 x + 49 x^{2}

= (1 + \frac{2}{3} x)^{7} (9 + 42 x + 49 x^{2})

Wende das Distributivgesetz an:

m (a + b + c) = ma + mb + m c

(1 + \frac{2}{3} x)^{7} (9 + 42 x + 49 x^{2}) = (1 + \frac{2}{3} x)^{7} \cdot 9 + (1 + \frac{2}{3} x)^{7} \cdot 42 x + (1 + \frac{2}{3} x)^{7} \cdot 49 x^{2}

= (1 + \frac{2}{3} x)^{7} \cdot 9 + (1 + \frac{2}{3} x)^{7} \cdot 42 x + (1 + \frac{2}{3} x)^{7} \cdot 49 x^{2}

= 9 (1 + \frac{2}{3} x)^{7} + 42 x (1 + \frac{2}{3} x)^{7} + 49 x^{2} (1 + \frac{2}{3} x)^{7}

Schreibe

9 (1 + \frac{2}{3} x)^{7}

um:

9 + 42 x + 84 x^{2} + \frac{280 x ^{3}}{3} + \frac{560 x ^{4}}{9} + \frac{224 x ^{5}}{9} + \frac{448 x ^{6}}{81} + \frac{128 x ^{7}}{243}

= 9 + 42 x + 84 x^{2} + \frac{280 x ^{3}}{3} + \frac{560 x ^{4}}{9} + \frac{224 x ^{5}}{9} + \frac{448 x ^{6}}{81} + \frac{128 x ^{7}}{243} + 42 x (1 + \frac{2}{3} x)^{7} + 49 x^{2} (1 + \frac{2}{3} x)^{7}

Schreibe

42 x (1 + \frac{2}{3} x)^{7}

um:

42 x + 196 x^{2} + 392 x^{3} + \frac{3920 x ^{4}}{9} + \frac{7840 x ^{5}}{27} + \frac{3136 x ^{6}}{27} + \frac{6272 x ^{7}}{243} + \frac{1792 x ^{8}}{729}

= 9 + 42 x + 84 x^{2} + \frac{280 x ^{3}}{3} + \frac{560 x ^{4}}{9} + \frac{224 x ^{5}}{9} + \frac{448 x ^{6}}{81} + \frac{128 x ^{7}}{243} + 42 x + 196 x^{2} + 392 x^{3} + \frac{3920 x ^{4}}{9} + \frac{7840 x ^{5}}{27} + \frac{3136 x ^{6}}{27} + \frac{6272 x ^{7}}{243} + \frac{1792 x ^{8}}{729} + 49 x^{2} (1 + \frac{2}{3} x)^{7}

Schreibe

49 x^{2} (1 + \frac{2}{3} x)^{7}

um:

49 x^{2} + \frac{686 x ^{3}}{3} + \frac{1372 x ^{4}}{3} + \frac{13720 x ^{5}}{27} + \frac{27440 x ^{6}}{81} + \frac{10976 x ^{7}}{81} + \frac{21952 x ^{8}}{729} + \frac{6272 x ^{9}}{2187}

= 9 + 42 x + 84 x^{2} + \frac{280 x ^{3}}{3} + \frac{560 x ^{4}}{9} + \frac{224 x ^{5}}{9} + \frac{448 x ^{6}}{81} + \frac{128 x ^{7}}{243} + 42 x + 196 x^{2} + 392 x^{3} + \frac{3920 x ^{4}}{9} + \frac{7840 x ^{5}}{27} + \frac{3136 x ^{6}}{27} + \frac{6272 x ^{7}}{243} + \frac{1792 x ^{8}}{729} + 49 x^{2} + \frac{686 x ^{3}}{3} + \frac{1372 x ^{4}}{3} + \frac{13720 x ^{5}}{27} + \frac{27440 x ^{6}}{81} + \frac{10976 x ^{7}}{81} + \frac{21952 x ^{8}}{729} + \frac{6272 x ^{9}}{2187}

Vereinfache

9 + 42 x + 84 x^{2} + \frac{280 x ^{3}}{3} + \frac{560 x ^{4}}{9} + \frac{224 x ^{5}}{9} + \frac{448 x ^{6}}{81} + \frac{128 x ^{7}}{243} + 42 x + 196 x^{2} + 392 x^{3} + \frac{3920 x ^{4}}{9} + \frac{7840 x ^{5}}{27} + \frac{3136 x ^{6}}{27} + \frac{6272 x ^{7}}{243} + \frac{1792 x ^{8}}{729} + 49 x^{2} + \frac{686 x ^{3}}{3} + \frac{1372 x ^{4}}{3} + \frac{13720 x ^{5}}{27} + \frac{27440 x ^{6}}{81} + \frac{10976 x ^{7}}{81} + \frac{21952 x ^{8}}{729} + \frac{6272 x ^{9}}{2187} : 9 + 84 x + 329 x^{2} + 714 x^{3} + \frac{8596 x ^{4}}{9} + \frac{22232 x ^{5}}{27} + \frac{4144 x ^{6}}{9} + \frac{39328 x ^{7}}{243} + \frac{23744 x ^{8}}{729} + \frac{6272 x ^{9}}{2187}

= 9 + 84 x + 329 x^{2} + 714 x^{3} + \frac{8596 x ^{4}}{9} + \frac{22232 x ^{5}}{27} + \frac{4144 x ^{6}}{9} + \frac{39328 x ^{7}}{243} + \frac{23744 x ^{8}}{729} + \frac{6272 x ^{9}}{2187}

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