Simplify (5-2i)(4-i)

$(5 - 2 i) (4 - i)$

Step 1

Expand $(5 - 2 i) (4 - i)$ using the FOIL Method.

Apply the distributive property.

$5 (4 - i) - 2 i (4 - i)$

Apply the distributive property.

$5 \cdot 4 + 5 (- i) - 2 i (4 - i)$

Apply the distributive property.

$5 \cdot 4 + 5 (- i) - 2 i \cdot 4 - 2 i (- i)$

$5 \cdot 4 + 5 (- i) - 2 i \cdot 4 - 2 i (- i)$

Step 2

Simplify and combine like terms.

Simplify each term.

Multiply $5$ by $4$ .

$20 + 5 (- i) - 2 i \cdot 4 - 2 i (- i)$

Multiply $- 1$ by $5$ .

$20 - 5 i - 2 i \cdot 4 - 2 i (- i)$

Multiply $4$ by $- 2$ .

$20 - 5 i - 8 i - 2 i (- i)$

Multiply $- 2 i (- i)$ .

Multiply $- 1$ by $- 2$ .

$20 - 5 i - 8 i + 2 i i$

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

$20 - 5 i - 8 i + 2 (i^{1} i)$

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

$20 - 5 i - 8 i + 2 (i^{1} i^{1})$

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

$20 - 5 i - 8 i + 2 i^{1 + 1}$

Add $1$ and $1$ .

$20 - 5 i - 8 i + 2 i^{2}$

$20 - 5 i - 8 i + 2 i^{2}$

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

$20 - 5 i - 8 i + 2 \cdot - 1$

Multiply $2$ by $- 1$ .

$20 - 5 i - 8 i - 2$

$20 - 5 i - 8 i - 2$

Subtract $2$ from $20$ .

$18 - 5 i - 8 i$

Subtract $8 i$ from $- 5 i$ .

$18 - 13 i$

$18 - 13 i$

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