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

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

Step 1

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

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

Step 2

Simplify and combine like terms.

Simplify each term.

Multiply $5$ by $2$ .

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

Multiply $- 1$ by $5$ .

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

Multiply $2$ by $3$ .

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

Multiply $3 i (- i)$ .

Multiply $- 1$ by $3$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

Multiply $- 3$ by $- 1$ .

$10 - 5 i + 6 i + 3$

$10 - 5 i + 6 i + 3$

Add $10$ and $3$ .

$13 - 5 i + 6 i$

Add $- 5 i$ and $6 i$ .

$13 + i$

$13 + i$

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