vereinfachen (x+iy)^6+(x+iy)^3-1

Lösung

vereinfachen

(x + i y)^{6} + (x + i y)^{3} - 1

(x^{6} - 15 x^{4} y^{2} + 15 x^{2} y^{4} - y^{6} + x^{3} - 3 x y^{2} - 1) + (6 x^{5} y - 20 x^{3} y^{3} + 6 x y^{5} + 3 x^{2} y - y^{3}) i

Schritte zur Lösung

(x + i y)^{6} + (x + i y)^{3} - 1

(x + i y)^{6} = x^{6} + 6 i x^{5} y - 15 x^{4} y^{2} - 20 i x^{3} y^{3} + 15 x^{2} y^{4} + 6 i x y^{5} - y^{6}

(x + i y)^{3} = x^{3} + 3 x^{2} y i - 3 x y^{2} - i y^{3}

= x^{6} + 6 i x^{5} y - 15 x^{4} y^{2} - 20 i x^{3} y^{3} + 15 x^{2} y^{4} + 6 i x y^{5} - y^{6} + x^{3} + 3 x^{2} y i - 3 x y^{2} - i y^{3} - 1

Rewrite in standard complex form:

(x^{6} - 15 x^{4} y^{2} + 15 x^{2} y^{4} - y^{6} + x^{3} - 3 x y^{2} - 1) + (6 x^{5} y - 20 x^{3} y^{3} + 6 x y^{5} + 3 x^{2} y - y^{3}) i

= (x^{6} - 15 x^{4} y^{2} + 15 x^{2} y^{4} - y^{6} + x^{3} - 3 x y^{2} - 1) + (6 x^{5} y - 20 x^{3} y^{3} + 6 x y^{5} + 3 x^{2} y - y^{3}) i

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

s im pl i f y (x - 2 - 5) (x - 2 + 5)

s im pl i f y (1 - 2 y i)^{2}

e x p an d π r^{2} (2 π r - 84)

e x p an d (- \frac{π L ^{2} ( 90 - arccos ( \frac{2}{L} ) )}{360} - 4)^{2}