Simplify (a^m+b^n)(a^m-b^n)

$(a^{m} + b^{n}) (a^{m} - b^{n})$

Step 1

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

Apply the distributive property.

$a^{m} (a^{m} - b^{n}) + b^{n} (a^{m} - b^{n})$

Apply the distributive property.

$a^{m} a^{m} + a^{m} (- b^{n}) + b^{n} (a^{m} - b^{n})$

Apply the distributive property.

$a^{m} a^{m} + a^{m} (- b^{n}) + b^{n} a^{m} + b^{n} (- b^{n})$

Step 2

Simplify terms.

Combine the opposite terms in $a^{m} a^{m} + a^{m} (- b^{n}) + b^{n} a^{m} + b^{n} (- b^{n})$ .

Reorder the factors in the terms $a^{m} (- b^{n})$ and $b^{n} a^{m}$ .

$a^{m} a^{m} - b^{n} a^{m} + b^{n} a^{m} + b^{n} (- b^{n})$

Add $- b^{n} a^{m}$ and $b^{n} a^{m}$ .

$a^{m} a^{m} + 0 + b^{n} (- b^{n})$

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

$a^{m} a^{m} + b^{n} (- b^{n})$

Simplify each term.

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

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

$a^{m + m} + b^{n} (- b^{n})$

Add $m$ and $m$ .

$a^{2 m} + b^{n} (- b^{n})$

Rewrite using the commutative property of multiplication.

$a^{2 m} - b^{n} b^{n}$

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

Move $b^{n}$ .

$a^{2 m} - (b^{n} b^{n})$

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

$a^{2 m} - b^{n + n}$

Add $n$ and $n$ .

$a^{2 m} - b^{2 n}$