simplify (1+4i)^3

Solution

simplify

(1 + 4 i)^{3}

- 47 - 52 i

Solution steps

(1 + 4 i)^{3}

Apply Perfect Cube Formula:

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

(1 + 4 i)^{3} = 1^{3} + 3 \cdot 1^{2} \cdot 4 i + 3 \cdot 1 \cdot (4 i)^{2} + (4 i)^{3}

= 1^{3} + 3 \cdot 1^{2} \cdot 4 i + 3 \cdot 1 \cdot (4 i)^{2} + (4 i)^{3}

Simplify

1^{3} + 3 \cdot 1^{2} \cdot 4 i + 3 \cdot 1 \cdot (4 i)^{2} + (4 i)^{3} : - 52 i - 47

= - 52 i - 47

Rewrite in standard complex form:

- 47 - 52 i

= - 47 - 52 i

\frac{1}{x + 3}

- \frac{2}{3} (2)

5 \times 1 0^{6}

31 \cdot 4