simplify (z+3i)^3

Solution

simplify

(z + 3 i)^{3}

(z^{3} - 27 z) + (9 z^{2} - 27) i

Solution steps

(z + 3 i)^{3}

Apply Perfect Cube Formula:

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

(z + 3 i)^{3} = z^{3} + 3 z^{2} \cdot 3 i + 3 z (3 i)^{2} + (3 i)^{3}

= z^{3} + 3 z^{2} \cdot 3 i + 3 z (3 i)^{2} + (3 i)^{3}

3 z^{2} \cdot 3 i = 9 z^{2} i

3 z (3 i)^{2} = - 27 z

(3 i)^{3} = - 27 i

= z^{3} + 9 z^{2} i - 27 z - 27 i

Rewrite in standard complex form:

(z^{3} - 27 z) + (9 z^{2} - 27) i

= (z^{3} - 27 z) + (9 z^{2} - 27) i

\frac{x ( 3 x + x ^{2} - 9 )}{6}

(e^{ia} + e^{- ia})^{2}

\frac{( 7 - y ) ^{3}}{3}

(a + b x + c x^{2} + d x^{3} + e x^{4} + f x^{5})^{2}

(6 x^{2} - 3 x + 5) d x