Simplify (1+2i)*(5-3i)

$(1 + 2 i) \cdot (5 - 3 i)$

Step 1

Expand $(1 + 2 i) (5 - 3 i)$ using the FOIL Method.

Apply the distributive property.

$1 (5 - 3 i) + 2 i (5 - 3 i)$

Apply the distributive property.

$1 \cdot 5 + 1 (- 3 i) + 2 i (5 - 3 i)$

Apply the distributive property.

$1 \cdot 5 + 1 (- 3 i) + 2 i \cdot 5 + 2 i (- 3 i)$

Step 2

Simplify and combine like terms.

Simplify each term.

Multiply $5$ by $1$ .

$5 + 1 (- 3 i) + 2 i \cdot 5 + 2 i (- 3 i)$

Multiply $- 3 i$ by $1$ .

$5 - 3 i + 2 i \cdot 5 + 2 i (- 3 i)$

Multiply $5$ by $2$ .

$5 - 3 i + 10 i + 2 i (- 3 i)$

Multiply $2 i (- 3 i)$ .

Multiply $- 3$ by $2$ .

$5 - 3 i + 10 i - 6 i i$

Raise $i$ to the power of $1$ .

$5 - 3 i + 10 i - 6 (i^{1} i)$

Raise $i$ to the power of $1$ .

$5 - 3 i + 10 i - 6 (i^{1} i^{1})$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$5 - 3 i + 10 i - 6 i^{1 + 1}$

Add $1$ and $1$ .

$5 - 3 i + 10 i - 6 i^{2}$

Rewrite $i^{2}$ as $- 1$ .

$5 - 3 i + 10 i - 6 \cdot - 1$

Multiply $- 6$ by $- 1$ .

$5 - 3 i + 10 i + 6$

Add $5$ and $6$ .

$11 - 3 i + 10 i$

Add $- 3 i$ and $10 i$ .

$11 + 7 i$