simplify i(x+yi)^4

Solution

simplify

i (x + y i)^{4}

(- 4 x^{3} y + 4 x y^{3}) + (x^{4} - 6 x^{2} y^{2} + y^{4}) i

Solution steps

i (x + y i)^{4}

Expand

(x + y i)^{4} : x^{4} + 4 i x^{3} y + 6 i^{2} x^{2} y^{2} + 4 i^{3} x y^{3} + i^{4} y^{4}

= i (x^{4} + 4 i x^{3} y + 6 i^{2} x^{2} y^{2} + 4 i^{3} x y^{3} + i^{4} y^{4})

Distribute parentheses

= i x^{4} + i 4 i x^{3} y + i 6 i^{2} x^{2} y^{2} + i 4 i^{3} x y^{3} + i i^{4} y^{4}

Simplify

i x^{4} + i 4 i x^{3} y + i 6 i^{2} x^{2} y^{2} + i 4 i^{3} x y^{3} + i i^{4} y^{4} : x^{4} i - 4 x^{3} y - 6 x^{2} i y^{2} + 4 x y^{3} + i y^{4}

= x^{4} i - 4 x^{3} y - 6 x^{2} i y^{2} + 4 x y^{3} + i y^{4}

Rewrite in standard complex form:

(- 4 x^{3} y + 4 x y^{3}) + (x^{4} - 6 x^{2} y^{2} + y^{4}) i

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

sin (2 t) \cdot x (t - π)

\frac{cos ^{2} ( x )}{( sin ( x ) + 1 ) ^{2}}

(1 - x^{\frac{2}{3}})^{3}

x (x - \frac{1}{2})

(9 x + 2) e^{4 x}

What is i(x+yi)^4 ?
The solution to i(x+yi)^4 is (-4x^3y+4xy^3)+(x^4-6x^2y^2+y^4)i