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

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

Step 1

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 2

Simplify and combine like terms.

Simplify each term.

Multiply $2$ by $2$ .

$4 + 2 (- 7 i) + 7 i \cdot 2 + 7 i (- 7 i)$

Multiply $- 7$ by $2$ .

$4 - 14 i + 7 i \cdot 2 + 7 i (- 7 i)$

Multiply $2$ by $7$ .

$4 - 14 i + 14 i + 7 i (- 7 i)$

Multiply $7 i (- 7 i)$ .

Multiply $- 7$ by $7$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

Multiply $- 49$ by $- 1$ .

$4 - 14 i + 14 i + 49$

Add $4$ and $49$ .

$53 - 14 i + 14 i$

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

$53 + 0$

Add $53$ and $0$ .

$53$