Simplify (4+7i)(4-7i)

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

Step 1

Expand $(4 + 7 i) (4 - 7 i)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 2

Simplify and combine like terms.

Simplify each term.

Multiply $4$ by $4$ .

$16 + 4 (- 7 i) + 7 i \cdot 4 + 7 i (- 7 i)$

Multiply $- 7$ by $4$ .

$16 - 28 i + 7 i \cdot 4 + 7 i (- 7 i)$

Multiply $4$ by $7$ .

$16 - 28 i + 28 i + 7 i (- 7 i)$

Multiply $7 i (- 7 i)$ .

Multiply $- 7$ by $7$ .

$16 - 28 i + 28 i - 49 i i$

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

$16 - 28 i + 28 i - 49 (i^{1} i)$

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

$16 - 28 i + 28 i - 49 (i^{1} i^{1})$

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

$16 - 28 i + 28 i - 49 i^{1 + 1}$

Add $1$ and $1$ .

$16 - 28 i + 28 i - 49 i^{2}$

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

$16 - 28 i + 28 i - 49 \cdot - 1$

Multiply $- 49$ by $- 1$ .

$16 - 28 i + 28 i + 49$

Add $16$ and $49$ .

$65 - 28 i + 28 i$

Add $- 28 i$ and $28 i$ .

$65 + 0$

Add $65$ and $0$ .

$65$