simplify (a+b*i)^3

Solution

simplify

(a + b \cdot i)^{3}

(a^{3} - 3 a b^{2}) + (3 a^{2} b - b^{3}) i

Solution steps

(a + bi)^{3}

Apply Perfect Cube Formula:

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

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

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

3 a (bi)^{2} = - 3 a b^{2}

(bi)^{3} = - i b^{3}

= a^{3} + 3 a^{2} bi - 3 a b^{2} - i b^{3}

Rewrite in standard complex form:

(a^{3} - 3 a b^{2}) + (3 a^{2} b - b^{3}) i

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

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

(y^{2} + \frac{1}{4 y ^{2}})^{2}

\frac{2}{9} (3 k - 1)

(4 x^{2} + x - \frac{3}{2}) (6 x - 8)