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}

Simplify

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

(x + 5) (x - 5)

(\frac{x}{2} - 5 y)^{4}

(1 + 3 i)^{4}

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

(x + y) (x + y)