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Complex Expressions in the form 𝑎+𝑏𝑖

• To simplify expressions with complex numbers, combine    like terms   . 
• When the square root of a    negative    term is present, simplify it out    first   .
• As you simplify and combine terms, remember the rules for the    imaginary unit   . 

Example 1

Simplify. Then classify by all sets to which it belongs: real, pure imaginary, complex.

$13\sqrt{–1}–2\sqrt{–81}$

$$\begin{gathered} 13i–2i\sqrt{9^2} \\ 13i–2i\left(9\right) \\ 13i–18i \\ –5i \\ \mathrm{pure} \mathrm{imaginary}, \mathrm{complex} \end{gathered}$$

Example 2

Simplify. Then classify by all sets to which it belongs: real, pure imaginary, complex.

$\left(4–3i\right)+2\left(5+7i\right)$

$$\begin{gathered} 4–3i+10+14i \\ 14+11i, \mathrm{complex} \end{gathered}$$

Example 3

Simplify.

$\left(2x–9i\right)\left(2x+9i\right)$

Example 4

Simplify.

${\left(4+i\right)}^2–3i^3\left(i+2\right)$

$$\begin{gathered} 16+8i+i^2–3i^4–6i^3 \\ 16+8i+\left(–1\right)–3\left(1\right)–6\left(–i\right) \\ 16–1–3+8i+6i \\ 12+14i \end{gathered}$$