Simplify (7+2i)(7-2i)

$(7 + 2 i) (7 - 2 i)$

Step 1

Expand $(7 + 2 i) (7 - 2 i)$ using the FOIL Method.

Apply the distributive property.

$7 (7 - 2 i) + 2 i (7 - 2 i)$

Apply the distributive property.

$7 \cdot 7 + 7 (- 2 i) + 2 i (7 - 2 i)$

Apply the distributive property.

$7 \cdot 7 + 7 (- 2 i) + 2 i \cdot 7 + 2 i (- 2 i)$

Step 2

Simplify and combine like terms.

Simplify each term.

Multiply $7$ by $7$ .

$49 + 7 (- 2 i) + 2 i \cdot 7 + 2 i (- 2 i)$

Multiply $- 2$ by $7$ .

$49 - 14 i + 2 i \cdot 7 + 2 i (- 2 i)$

Multiply $7$ by $2$ .

$49 - 14 i + 14 i + 2 i (- 2 i)$

Multiply $2 i (- 2 i)$ .

Multiply $- 2$ by $2$ .

$49 - 14 i + 14 i - 4 i i$

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

$49 - 14 i + 14 i - 4 (i^{1} i)$

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

$49 - 14 i + 14 i - 4 (i^{1} i^{1})$

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

$49 - 14 i + 14 i - 4 i^{1 + 1}$

Add $1$ and $1$ .

$49 - 14 i + 14 i - 4 i^{2}$

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

$49 - 14 i + 14 i - 4 \cdot - 1$

Multiply $- 4$ by $- 1$ .

$49 - 14 i + 14 i + 4$

Add $49$ and $4$ .

$53 - 14 i + 14 i$

Add $- 14 i$ and $14 i$ .

$53 + 0$

Add $53$ and $0$ .

$53$