$$ (3+3i)^4 = (3+3i)^2 \cdot (3+3i)^2 $$

Find $ \left(3+3i\right)^2 $ using formula.

$$ (A + B)^2 = \color{blue}{A^2} + 2 \cdot A \cdot B + \color{red}{B^2} $$

where $ A = \color{blue}{ 3 } $ and $ B = \color{red}{ 3i }$.

$$ \begin{aligned}\left(3+3i\right)^2 = \color{blue}{3^2} +2 \cdot 3 \cdot 3i + \color{red}{\left( 3i \right)^2} = 9+18i+9i^2\end{aligned} $$

Multiply each term of $ \left( \color{blue}{9+18i+9i^2}\right) $ by each term in $ \left( 9+18i+9i^2\right) $.

$$ \left( \color{blue}{9+18i+9i^2}\right) \cdot \left( 9+18i+9i^2\right) = \\ = 81+162i+81i^2+162i+324i^2+162i^3+81i^2+162i^3+81i^4 $$

Combine like terms:

$$ 81+ \color{blue}{162i} + \color{red}{81i^2} + \color{blue}{162i} + \color{green}{324i^2} + \color{orange}{162i^3} + \color{green}{81i^2} + \color{orange}{162i^3} +81i^4 = \\ = 81i^4+ \color{orange}{324i^3} + \color{green}{486i^2} + \color{blue}{324i} +81 $$162i^3+162i^3=324i^3

Combine like terms:

$$ 81i^4+324i^3+486i^2+324i+ \, \color{blue}{ \cancel{81}} \, \, \color{blue}{ -\cancel{81}} \, = 81i^4+324i^3+486i^2+324i $$

$$ 81i^4 = 81 \cdot i^2 \cdot i^2 = 81 \cdot ( - 1) \cdot ( - 1) = 81 $$

$$ 324i^3 = 324 \cdot \color{blue}{i^2} \cdot i = 324 \cdot ( \color{blue}{-1}) \cdot i = -324 \cdot \, i $$

$$ 486i^2 = 486 \cdot (-1) = -486 $$

Combine like terms:

$$ \color{blue}{81} \, \color{red}{ -\cancel{324i}} \, \color{blue}{-486} + \, \color{red}{ \cancel{324i}} \, = \color{blue}{-405} $$

Multiply $ \dfrac{1}{4} $ by $ -405 $ to get $ \dfrac{-405}{4} $.

Write $ -405 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: Multiply numerators and denominators.

$$ \begin{aligned} \frac{1}{4} \cdot -405 = \frac{1}{4} \cdot \frac{-405}{\color{red}{1}} = \frac{-405}{4} \end{aligned} $$