Simplify (x^2-y^2)^2

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

Step 1

Rewrite ${(x^{2} - y^{2})}^{2}$ as $(x^{2} - y^{2}) (x^{2} - y^{2})$ .

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

Step 2

Expand $(x^{2} - y^{2}) (x^{2} - y^{2})$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

$x^{2} x^{2} + x^{2} (- y^{2}) - y^{2} (x^{2} - y^{2})$

Apply the distributive property.

$x^{2} x^{2} + x^{2} (- y^{2}) - y^{2} x^{2} - y^{2} (- y^{2})$

Step 3

Simplify and combine like terms.

Simplify each term.

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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

$x^{2 + 2} + x^{2} (- y^{2}) - y^{2} x^{2} - y^{2} (- y^{2})$

Add $2$ and $2$ .

$x^{4} + x^{2} (- y^{2}) - y^{2} x^{2} - y^{2} (- y^{2})$

Rewrite using the commutative property of multiplication.

$x^{4} - x^{2} y^{2} - y^{2} x^{2} - y^{2} (- y^{2})$

Rewrite using the commutative property of multiplication.

$x^{4} - x^{2} y^{2} - y^{2} x^{2} - 1 \cdot - 1 y^{2} y^{2}$

Multiply $y^{2}$ by $y^{2}$ by adding the exponents.

Move $y^{2}$ .

$x^{4} - x^{2} y^{2} - y^{2} x^{2} - 1 \cdot - 1 (y^{2} y^{2})$

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

$x^{4} - x^{2} y^{2} - y^{2} x^{2} - 1 \cdot - 1 y^{2 + 2}$

Add $2$ and $2$ .

$x^{4} - x^{2} y^{2} - y^{2} x^{2} - 1 \cdot - 1 y^{4}$

Multiply $- 1$ by $- 1$ .

$x^{4} - x^{2} y^{2} - y^{2} x^{2} + 1 y^{4}$

Multiply $y^{4}$ by $1$ .

$x^{4} - x^{2} y^{2} - y^{2} x^{2} + y^{4}$

Subtract $y^{2} x^{2}$ from $- x^{2} y^{2}$ .

Move $y^{2}$ .

$x^{4} - x^{2} y^{2} - 1 x^{2} y^{2} + y^{4}$

Subtract $x^{2} y^{2}$ from $- x^{2} y^{2}$ .

$x^{4} - 2 x^{2} y^{2} + y^{4}$