simplify (a+ib)^3

Solution

simplify

(a + ib)^{3}

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

Solution steps

(a + ib)^{3}

Apply Perfect Cube Formula:

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

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

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

3 a^{2} ib = 3 a^{2} bi

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

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

(2 + 3 i)^{5}

\frac{1}{( s + 1 ) ^{3}}

x^{\frac{1}{7}} (x + 8)

(1 + tan (α) tan (β)) tan (α - β)