Simplify (1+i)(2+3i)(4-3i)

$(1 + i) (2 + 3 i) (4 - 3 i)$

Step 1

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

Apply the distributive property.

$(1 (2 + 3 i) + i (2 + 3 i)) (4 - 3 i)$

Apply the distributive property.

$(1 \cdot 2 + 1 (3 i) + i (2 + 3 i)) (4 - 3 i)$

Apply the distributive property.

$(1 \cdot 2 + 1 (3 i) + i \cdot 2 + i (3 i)) (4 - 3 i)$

Step 2

Simplify and combine like terms.

Simplify each term.

Multiply $2$ by $1$ .

$(2 + 1 (3 i) + i \cdot 2 + i (3 i)) (4 - 3 i)$

Multiply $3 i$ by $1$ .

$(2 + 3 i + i \cdot 2 + i (3 i)) (4 - 3 i)$

Move $2$ to the left of $i$ .

$(2 + 3 i + 2 \cdot i + i (3 i)) (4 - 3 i)$

Multiply $i (3 i)$ .

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

$(2 + 3 i + 2 i + 3 (i^{1} i)) (4 - 3 i)$

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

$(2 + 3 i + 2 i + 3 (i^{1} i^{1})) (4 - 3 i)$

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

$(2 + 3 i + 2 i + 3 i^{1 + 1}) (4 - 3 i)$

Add $1$ and $1$ .

$(2 + 3 i + 2 i + 3 i^{2}) (4 - 3 i)$

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

$(2 + 3 i + 2 i + 3 \cdot - 1) (4 - 3 i)$

Multiply $3$ by $- 1$ .

$(2 + 3 i + 2 i - 3) (4 - 3 i)$

Subtract $3$ from $2$ .

$(- 1 + 3 i + 2 i) (4 - 3 i)$

Add $3 i$ and $2 i$ .

$(- 1 + 5 i) (4 - 3 i)$

Step 3

Expand $(- 1 + 5 i) (4 - 3 i)$ using the FOIL Method.

Apply the distributive property.

$- 1 (4 - 3 i) + 5 i (4 - 3 i)$

Apply the distributive property.

$- 1 \cdot 4 - 1 (- 3 i) + 5 i (4 - 3 i)$

Apply the distributive property.

$- 1 \cdot 4 - 1 (- 3 i) + 5 i \cdot 4 + 5 i (- 3 i)$

Step 4

Simplify and combine like terms.

Simplify each term.

Multiply $- 1$ by $4$ .

$- 4 - 1 (- 3 i) + 5 i \cdot 4 + 5 i (- 3 i)$

Multiply $- 3$ by $- 1$ .

$- 4 + 3 i + 5 i \cdot 4 + 5 i (- 3 i)$

Multiply $4$ by $5$ .

$- 4 + 3 i + 20 i + 5 i (- 3 i)$

Multiply $5 i (- 3 i)$ .

Multiply $- 3$ by $5$ .

$- 4 + 3 i + 20 i - 15 i i$

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

$- 4 + 3 i + 20 i - 15 (i^{1} i)$

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

$- 4 + 3 i + 20 i - 15 (i^{1} i^{1})$

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

$- 4 + 3 i + 20 i - 15 i^{1 + 1}$

Add $1$ and $1$ .

$- 4 + 3 i + 20 i - 15 i^{2}$

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

$- 4 + 3 i + 20 i - 15 \cdot - 1$

Multiply $- 15$ by $- 1$ .

$- 4 + 3 i + 20 i + 15$

Add $- 4$ and $15$ .

$11 + 3 i + 20 i$

Add $3 i$ and $20 i$ .

$11 + 23 i$

Simplify (1+i)(2+3i)(4-3i)

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