Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

Apply the distributive property.

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

Step 1.2.2

Apply the distributive property.

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

Step 1.2.3

Apply the distributive property.

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

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

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

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Step 1.3.1.1.1

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

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

Step 1.3.1.1.2

Add $2$ and $2$ .

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

Step 1.3.1.2

Rewrite using the commutative property of multiplication.

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

Step 1.3.1.3

Rewrite using the commutative property of multiplication.

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

Step 1.3.1.4

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

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Step 1.3.1.4.1

Move $y^{2}$ .

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

Step 1.3.1.4.2

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

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

Step 1.3.1.4.3

Add $2$ and $2$ .

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

Step 1.3.1.5

Multiply $- 1$ by $- 1$ .

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

Step 1.3.1.6

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

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

Step 1.3.2

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

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Step 1.3.2.1

Move $y^{2}$ .

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

Step 1.3.2.2

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

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

Step 1.4

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 1.4.1

Apply the product rule to $2 x y$ .

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

Step 1.4.2

Apply the product rule to $2 x$ .

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

Step 1.5

Raise $2$ to the power of $2$ .

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

Step 2

Add $- 2 x^{2} y^{2}$ and $4 x^{2} y^{2}$ .

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