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 $- b i a$ .

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

Subtract $i a b$ from $i a b$ .

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

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 $b$ .

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

Multiply $b$ by $b$ .

$a^{2} - b^{2} i i$

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 $- 1$ .

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

Multiply $- b^{2} \cdot - 1$ .

Multiply $- 1$ by $- 1$ .

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

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

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