simplify i(x+iy)^3

Solution

simplify

i (x + i y)^{3}

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

Solution steps

i (x + i y)^{3}

Expand

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

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

Distribute parentheses

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

i x^{3} = x^{3} i

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

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

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

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

Rewrite in standard complex form:

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

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

(sin (θ) + 3) (sin (θ) - 4)

\frac{( 2 x + 5 ) ( 2 x + 1 )}{3 x ( 2 x - 1 )}

(x^{4})^{2}

(\frac{x ^{2}}{2} - \frac{2}{x})^{9}