simplify (x+i*y)^3

Solution

simplify

(x + i \cdot 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

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