Simplify (a+bi)(a-bi)

$(a + b i) (a - b i)$

Step 1

Expand $(a + b i) (a - b i)$ using the FOIL Method.

Apply the distributive property.

$a (a - b i) + b i (a - b i)$

Apply the distributive property.

$a \cdot a + a (- b i) + b i (a - b i)$

Apply the distributive property.

$a \cdot a + a (- b i) + b i a + b i (- b i)$

Step 2

Simplify terms.

Combine the opposite terms in $a \cdot a + a (- b i) + b i a + b i (- b i)$ .

Reorder the factors in the terms $a (- b i)$ and $b i a$ .

$a \cdot a - i a b + i a b + b i (- b i)$

Add $- i a b$ and $i a b$ .

$a \cdot a + 0 + b i (- b i)$

Add $a \cdot a$ and $0$ .

$a \cdot a + b i (- b i)$

Simplify each term.

Multiply $a$ by $a$ .

$a^{2} + b i (- b i)$

Multiply $b$ by $b$ by adding the exponents.

Move $b$ .

$a^{2} + b \cdot b i (- 1 i)$

Multiply $b$ by $b$ .

$a^{2} + b^{2} i (- 1 i)$

Rewrite $- 1 i$ as $- i$ .

$a^{2} + b^{2} i (- i)$

Multiply $b^{2} i (- i)$ .

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

$a^{2} + b^{2} (- (i^{1} i))$

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

$a^{2} + b^{2} (- (i^{1} i^{1}))$

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

$a^{2} + b^{2} (- i^{1 + 1})$

Add $1$ and $1$ .

$a^{2} + b^{2} (- i^{2})$

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

$a^{2} + b^{2} (- - 1)$

Multiply $- 1$ by $- 1$ .

$a^{2} + b^{2} \cdot 1$

Multiply $b^{2}$ by $1$ .

$a^{2} + b^{2}$