simplify (x+iy)^3

Solution

simplify

(x + i y)^{3}

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

Solution steps

(x + i y)^{3}

Apply Perfect Cube Formula:

(a + b)^{3} = a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}

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

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

3 x^{2} i y = 3 x^{2} y i

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

(i y)^{3} = - i y^{3}

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

Rewrite in standard complex form:

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

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

(e^{x} + e^{- x})^{2}

(1 - cos (x))^{2}

(5 d + 8 e) (5 d - 8 e)

(a + bi)^{2}

(- 2 + i)^{4}