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

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

Step 1

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 2

Simplify and combine like terms.

Simplify each term.

Multiply $5$ by $1$ .

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

Multiply $3$ by $5$ .

$5 + 15 i + 2 i \cdot 1 + 2 i (3 i)$

Multiply $2$ by $1$ .

$5 + 15 i + 2 i + 2 i (3 i)$

Multiply $2 i (3 i)$ .

Multiply $3$ by $2$ .

$5 + 15 i + 2 i + 6 i i$

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

$5 + 15 i + 2 i + 6 (i^{1} i)$

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

$5 + 15 i + 2 i + 6 (i^{1} i^{1})$

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

$5 + 15 i + 2 i + 6 i^{1 + 1}$

Add $1$ and $1$ .

$5 + 15 i + 2 i + 6 i^{2}$

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

$5 + 15 i + 2 i + 6 \cdot - 1$

Multiply $6$ by $- 1$ .

$5 + 15 i + 2 i - 6$

Subtract $6$ from $5$ .

$- 1 + 15 i + 2 i$

Add $15 i$ and $2 i$ .

$- 1 + 17 i$