simplify (a+i)^3

Solution

simplify

(a + i)^{3}

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

Solution steps

(a + i)^{3}

Apply Perfect Cube Formula:

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

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

= a^{3} + 3 a^{2} i + 3 a i^{2} + i^{3}

3 a i^{2} = - 3 a

i^{3} = - i

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

Rewrite in standard complex form:

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

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

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

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

(x + 2) (x - 1 + 7) (x - 1 - 7)

(1 - \frac{x}{10})^{6}

\frac{s}{( s ^{2} + 1 ) ( s - 1 )}