simplify (n+2i)^3

Solution

simplify

(n + 2 i)^{3}

(n^{3} - 12 n) + (6 n^{2} - 8) i

Solution steps

(n + 2 i)^{3}

Apply Perfect Cube Formula:

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

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

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

3 n^{2} \cdot 2 i = 6 n^{2} i

3 n (2 i)^{2} = - 12 n

(2 i)^{3} = - 8 i

= n^{3} + 6 n^{2} i - 12 n - 8 i

Rewrite in standard complex form:

(n^{3} - 12 n) + (6 n^{2} - 8) i

= (n^{3} - 12 n) + (6 n^{2} - 8) i

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

(x^{2} + \frac{1}{x ^{2}})^{5}

sin (\frac{π}{3} - x)

(3 x - 1) (x)

(12 + 3 x) 2