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}

Simplify

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

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

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

y^{\frac{1}{3}} (y^{\frac{2}{3}} + y^{\frac{5}{3}})

(x - 32) (x + 32)

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

B (5, - \frac{π}{3})

(2 x - \frac{1}{x})^{4}