simplify (a+bi)^3

Solution

simplify

(a + bi)^{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

(1 + sin (x))^{2}

(1 - i)^{5}

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

(x + 2)^{2}

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