Find the Product (a^x-b^y)(a^x+b^y)

$(a^{x} - b^{y}) (a^{x} + b^{y})$

Step 1

Expand $(a^{x} - b^{y}) (a^{x} + b^{y})$ using the FOIL Method.

Apply the distributive property.

$a^{x} (a^{x} + b^{y}) - b^{y} (a^{x} + b^{y})$

Apply the distributive property.

$a^{x} a^{x} + a^{x} b^{y} - b^{y} (a^{x} + b^{y})$

Apply the distributive property.

$a^{x} a^{x} + a^{x} b^{y} - b^{y} a^{x} - b^{y} b^{y}$

Step 2

Simplify terms.

Combine the opposite terms in $a^{x} a^{x} + a^{x} b^{y} - b^{y} a^{x} - b^{y} b^{y}$ .

Reorder the factors in the terms $a^{x} b^{y}$ and $- b^{y} a^{x}$ .

$a^{x} a^{x} + b^{y} a^{x} - b^{y} a^{x} - b^{y} b^{y}$

Subtract $b^{y} a^{x}$ from $b^{y} a^{x}$ .

$a^{x} a^{x} + 0 - b^{y} b^{y}$

Add $a^{x} a^{x}$ and $0$ .

$a^{x} a^{x} - b^{y} b^{y}$

Simplify each term.

Multiply $a^{x}$ by $a^{x}$ by adding the exponents.

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

$a^{x + x} - b^{y} b^{y}$

Add $x$ and $x$ .

$a^{2 x} - b^{y} b^{y}$

Multiply $b^{y}$ by $b^{y}$ by adding the exponents.

Move $b^{y}$ .

$a^{2 x} - (b^{y} b^{y})$

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

$a^{2 x} - b^{y + y}$

Add $y$ and $y$ .

$a^{2 x} - b^{2 y}$