simplify (z-i)^3

Solution

simplify

(z - i)^{3}

(z^{3} - 3 z) + (- 3 z^{2} + 1) i

Solution steps

(z - i)^{3}

Apply Perfect Cube Formula:

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

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

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

Simplify

z^{3} - 3 z^{2} i + 3 z i^{2} - i^{3} : z^{3} - 3 z^{2} i - 3 z + i

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

Rewrite in standard complex form:

(z^{3} - 3 z) + (- 3 z^{2} + 1) i

= (z^{3} - 3 z) + (- 3 z^{2} + 1) i

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

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

(sin (x) - tan (x)) (cos (x) - cot (x))

\frac{4 x + 3}{( x - 2 ) ( x + 1 ) ^{2}}

(3 x^{2} - \frac{2}{x})^{5}