Finite Math Examples

${(2 x^{2} - \frac{1}{400} y^{2})}^{2} - {(2 x^{2} + \frac{1}{400} y^{2})}^{2}$

Step 1

Simplify each term.

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Step 1.1

Combine $y^{2}$ and $\frac{1}{400}$ .

${(2 x^{2} - \frac{y^{2}}{400})}^{2} - {(2 x^{2} + \frac{1}{400} y^{2})}^{2}$

Step 1.2

Rewrite ${(2 x^{2} - \frac{y^{2}}{400})}^{2}$ as $(2 x^{2} - \frac{y^{2}}{400}) (2 x^{2} - \frac{y^{2}}{400})$ .

$(2 x^{2} - \frac{y^{2}}{400}) (2 x^{2} - \frac{y^{2}}{400}) - {(2 x^{2} + \frac{1}{400} y^{2})}^{2}$

Step 1.3

Expand $(2 x^{2} - \frac{y^{2}}{400}) (2 x^{2} - \frac{y^{2}}{400})$ using the FOIL Method.

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Step 1.3.1

Apply the distributive property.

$2 x^{2} (2 x^{2} - \frac{y^{2}}{400}) - \frac{y^{2}}{400} (2 x^{2} - \frac{y^{2}}{400}) - {(2 x^{2} + \frac{1}{400} y^{2})}^{2}$

Step 1.3.2

Apply the distributive property.

$2 x^{2} (2 x^{2}) + 2 x^{2} (- \frac{y^{2}}{400}) - \frac{y^{2}}{400} (2 x^{2} - \frac{y^{2}}{400}) - {(2 x^{2} + \frac{1}{400} y^{2})}^{2}$

Step 1.3.3

Apply the distributive property.

$2 x^{2} (2 x^{2}) + 2 x^{2} (- \frac{y^{2}}{400}) - \frac{y^{2}}{400} (2 x^{2}) - \frac{y^{2}}{400} (- \frac{y^{2}}{400}) - {(2 x^{2} + \frac{1}{400} y^{2})}^{2}$

Step 1.4

Simplify and combine like terms.

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Step 1.4.1

Simplify each term.

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Step 1.4.1.1

Rewrite using the commutative property of multiplication.

$2 \cdot 2 x^{2} x^{2} + 2 x^{2} (- \frac{y^{2}}{400}) - \frac{y^{2}}{400} (2 x^{2}) - \frac{y^{2}}{400} (- \frac{y^{2}}{400}) - {(2 x^{2} + \frac{1}{400} y^{2})}^{2}$

Step 1.4.1.2

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

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Step 1.4.1.2.1

Move $x^{2}$ .

$2 \cdot 2 (x^{2} x^{2}) + 2 x^{2} (- \frac{y^{2}}{400}) - \frac{y^{2}}{400} (2 x^{2}) - \frac{y^{2}}{400} (- \frac{y^{2}}{400}) - {(2 x^{2} + \frac{1}{400} y^{2})}^{2}$

Step 1.4.1.2.2

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

$2 \cdot 2 x^{2 + 2} + 2 x^{2} (- \frac{y^{2}}{400}) - \frac{y^{2}}{400} (2 x^{2}) - \frac{y^{2}}{400} (- \frac{y^{2}}{400}) - {(2 x^{2} + \frac{1}{400} y^{2})}^{2}$

Step 1.4.1.2.3

Add $2$ and $2$ .

$2 \cdot 2 x^{4} + 2 x^{2} (- \frac{y^{2}}{400}) - \frac{y^{2}}{400} (2 x^{2}) - \frac{y^{2}}{400} (- \frac{y^{2}}{400}) - {(2 x^{2} + \frac{1}{400} y^{2})}^{2}$

Step 1.4.1.3

Multiply $2$ by $2$ .

$4 x^{4} + 2 x^{2} (- \frac{y^{2}}{400}) - \frac{y^{2}}{400} (2 x^{2}) - \frac{y^{2}}{400} (- \frac{y^{2}}{400}) - {(2 x^{2} + \frac{1}{400} y^{2})}^{2}$

Step 1.4.1.4

Cancel the common factor of $2$ .

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Step 1.4.1.4.1

Move the leading negative in $- \frac{y^{2}}{400}$ into the numerator.

$4 x^{4} + 2 x^{2} \frac{- y^{2}}{400} - \frac{y^{2}}{400} (2 x^{2}) - \frac{y^{2}}{400} (- \frac{y^{2}}{400}) - {(2 x^{2} + \frac{1}{400} y^{2})}^{2}$

Step 1.4.1.4.2

Factor $2$ out of $2 x^{2}$ .

$4 x^{4} + 2 (x^{2}) \frac{- y^{2}}{400} - \frac{y^{2}}{400} (2 x^{2}) - \frac{y^{2}}{400} (- \frac{y^{2}}{400}) - {(2 x^{2} + \frac{1}{400} y^{2})}^{2}$

Step 1.4.1.4.3

Factor $2$ out of $400$ .

$4 x^{4} + 2 (x^{2}) \frac{- y^{2}}{2 (200)} - \frac{y^{2}}{400} (2 x^{2}) - \frac{y^{2}}{400} (- \frac{y^{2}}{400}) - {(2 x^{2} + \frac{1}{400} y^{2})}^{2}$

Step 1.4.1.4.4

Cancel the common factor.

$4 x^{4} + 2 x^{2} \frac{- y^{2}}{2 \cdot 200} - \frac{y^{2}}{400} (2 x^{2}) - \frac{y^{2}}{400} (- \frac{y^{2}}{400}) - {(2 x^{2} + \frac{1}{400} y^{2})}^{2}$

Step 1.4.1.4.5

Rewrite the expression.

$4 x^{4} + x^{2} \frac{- y^{2}}{200} - \frac{y^{2}}{400} (2 x^{2}) - \frac{y^{2}}{400} (- \frac{y^{2}}{400}) - {(2 x^{2} + \frac{1}{400} y^{2})}^{2}$

Step 1.4.1.5

Combine $x^{2}$ and $\frac{- y^{2}}{200}$ .

$4 x^{4} + \frac{x^{2} (- y^{2})}{200} - \frac{y^{2}}{400} (2 x^{2}) - \frac{y^{2}}{400} (- \frac{y^{2}}{400}) - {(2 x^{2} + \frac{1}{400} y^{2})}^{2}$

Step 1.4.1.6

Factor out negative.

$4 x^{4} + \frac{- x^{2} y^{2}}{200} - \frac{y^{2}}{400} (2 x^{2}) - \frac{y^{2}}{400} (- \frac{y^{2}}{400}) - {(2 x^{2} + \frac{1}{400} y^{2})}^{2}$

Step 1.4.1.7

Move the negative in front of the fraction.

$4 x^{4} - \frac{x^{2} y^{2}}{200} - \frac{y^{2}}{400} (2 x^{2}) - \frac{y^{2}}{400} (- \frac{y^{2}}{400}) - {(2 x^{2} + \frac{1}{400} y^{2})}^{2}$

Step 1.4.1.8

Cancel the common factor of $2$ .

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Step 1.4.1.8.1

Move the leading negative in $- \frac{y^{2}}{400}$ into the numerator.

$4 x^{4} - \frac{x^{2} y^{2}}{200} + \frac{- y^{2}}{400} (2 x^{2}) - \frac{y^{2}}{400} (- \frac{y^{2}}{400}) - {(2 x^{2} + \frac{1}{400} y^{2})}^{2}$

Step 1.4.1.8.2

Factor $2$ out of $400$ .

$4 x^{4} - \frac{x^{2} y^{2}}{200} + \frac{- y^{2}}{2 (200)} (2 x^{2}) - \frac{y^{2}}{400} (- \frac{y^{2}}{400}) - {(2 x^{2} + \frac{1}{400} y^{2})}^{2}$

Step 1.4.1.8.3

Factor $2$ out of $2 x^{2}$ .

$4 x^{4} - \frac{x^{2} y^{2}}{200} + \frac{- y^{2}}{2 (200)} (2 (x^{2})) - \frac{y^{2}}{400} (- \frac{y^{2}}{400}) - {(2 x^{2} + \frac{1}{400} y^{2})}^{2}$

Step 1.4.1.8.4

Cancel the common factor.

$4 x^{4} - \frac{x^{2} y^{2}}{200} + \frac{- y^{2}}{2 \cdot 200} (2 x^{2}) - \frac{y^{2}}{400} (- \frac{y^{2}}{400}) - {(2 x^{2} + \frac{1}{400} y^{2})}^{2}$

Step 1.4.1.8.5

Rewrite the expression.

$4 x^{4} - \frac{x^{2} y^{2}}{200} + \frac{- y^{2}}{200} x^{2} - \frac{y^{2}}{400} (- \frac{y^{2}}{400}) - {(2 x^{2} + \frac{1}{400} y^{2})}^{2}$

Step 1.4.1.9

Combine $\frac{- y^{2}}{200}$ and $x^{2}$ .

$4 x^{4} - \frac{x^{2} y^{2}}{200} + \frac{- y^{2} x^{2}}{200} - \frac{y^{2}}{400} (- \frac{y^{2}}{400}) - {(2 x^{2} + \frac{1}{400} y^{2})}^{2}$

Step 1.4.1.10

Move the negative in front of the fraction.

$4 x^{4} - \frac{x^{2} y^{2}}{200} - \frac{y^{2} x^{2}}{200} - \frac{y^{2}}{400} (- \frac{y^{2}}{400}) - {(2 x^{2} + \frac{1}{400} y^{2})}^{2}$

Step 1.4.1.11

Multiply $- \frac{y^{2}}{400} (- \frac{y^{2}}{400})$ .

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Step 1.4.1.11.1

Multiply $- 1$ by $- 1$ .

$4 x^{4} - \frac{x^{2} y^{2}}{200} - \frac{y^{2} x^{2}}{200} + 1 \frac{y^{2}}{400} \frac{y^{2}}{400} - {(2 x^{2} + \frac{1}{400} y^{2})}^{2}$

Step 1.4.1.11.2

Multiply $\frac{y^{2}}{400}$ by $1$ .

$4 x^{4} - \frac{x^{2} y^{2}}{200} - \frac{y^{2} x^{2}}{200} + \frac{y^{2}}{400} \cdot \frac{y^{2}}{400} - {(2 x^{2} + \frac{1}{400} y^{2})}^{2}$

Step 1.4.1.11.3

Multiply $\frac{y^{2}}{400}$ by $\frac{y^{2}}{400}$ .

$4 x^{4} - \frac{x^{2} y^{2}}{200} - \frac{y^{2} x^{2}}{200} + \frac{y^{2} y^{2}}{400 \cdot 400} - {(2 x^{2} + \frac{1}{400} y^{2})}^{2}$

Step 1.4.1.11.4

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

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Step 1.4.1.11.4.1

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

$4 x^{4} - \frac{x^{2} y^{2}}{200} - \frac{y^{2} x^{2}}{200} + \frac{y^{2 + 2}}{400 \cdot 400} - {(2 x^{2} + \frac{1}{400} y^{2})}^{2}$

Step 1.4.1.11.4.2

Add $2$ and $2$ .

$4 x^{4} - \frac{x^{2} y^{2}}{200} - \frac{y^{2} x^{2}}{200} + \frac{y^{4}}{400 \cdot 400} - {(2 x^{2} + \frac{1}{400} y^{2})}^{2}$

Step 1.4.1.11.5

Multiply $400$ by $400$ .

$4 x^{4} - \frac{x^{2} y^{2}}{200} - \frac{y^{2} x^{2}}{200} + \frac{y^{4}}{160000} - {(2 x^{2} + \frac{1}{400} y^{2})}^{2}$

Step 1.4.2

Subtract $\frac{y^{2} x^{2}}{200}$ from $- \frac{x^{2} y^{2}}{200}$ .

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Step 1.4.2.1

Reorder $y^{2}$ and $x^{2}$ .

$4 x^{4} - \frac{x^{2} y^{2}}{200} - \frac{x^{2} y^{2}}{200} + \frac{y^{4}}{160000} - {(2 x^{2} + \frac{1}{400} y^{2})}^{2}$

Step 1.4.2.2

Subtract $\frac{x^{2} y^{2}}{200}$ from $- \frac{x^{2} y^{2}}{200}$ .

$4 x^{4} - 2 \frac{x^{2} y^{2}}{200} + \frac{y^{4}}{160000} - {(2 x^{2} + \frac{1}{400} y^{2})}^{2}$

Step 1.5

Simplify each term.

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Step 1.5.1

Cancel the common factor of $2$ .

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Step 1.5.1.1

Factor $2$ out of $- 2$ .

$4 x^{4} + 2 (- 1) \frac{x^{2} y^{2}}{200} + \frac{y^{4}}{160000} - {(2 x^{2} + \frac{1}{400} y^{2})}^{2}$

Step 1.5.1.2

Factor $2$ out of $200$ .

$4 x^{4} + 2 \cdot - 1 \frac{x^{2} y^{2}}{2 \cdot 100} + \frac{y^{4}}{160000} - {(2 x^{2} + \frac{1}{400} y^{2})}^{2}$

Step 1.5.1.3

Cancel the common factor.

$4 x^{4} + 2 \cdot - 1 \frac{x^{2} y^{2}}{2 \cdot 100} + \frac{y^{4}}{160000} - {(2 x^{2} + \frac{1}{400} y^{2})}^{2}$

Step 1.5.1.4

Rewrite the expression.

$4 x^{4} - 1 \frac{x^{2} y^{2}}{100} + \frac{y^{4}}{160000} - {(2 x^{2} + \frac{1}{400} y^{2})}^{2}$

Step 1.5.2

Rewrite $- 1 \frac{x^{2} y^{2}}{100}$ as $- \frac{x^{2} y^{2}}{100}$ .

$4 x^{4} - \frac{x^{2} y^{2}}{100} + \frac{y^{4}}{160000} - {(2 x^{2} + \frac{1}{400} y^{2})}^{2}$

Step 1.6

Combine $\frac{1}{400}$ and $y^{2}$ .

$4 x^{4} - \frac{x^{2} y^{2}}{100} + \frac{y^{4}}{160000} - {(2 x^{2} + \frac{y^{2}}{400})}^{2}$

Step 1.7

Rewrite ${(2 x^{2} + \frac{y^{2}}{400})}^{2}$ as $(2 x^{2} + \frac{y^{2}}{400}) (2 x^{2} + \frac{y^{2}}{400})$ .

$4 x^{4} - \frac{x^{2} y^{2}}{100} + \frac{y^{4}}{160000} - ((2 x^{2} + \frac{y^{2}}{400}) (2 x^{2} + \frac{y^{2}}{400}))$

Step 1.8

Expand $(2 x^{2} + \frac{y^{2}}{400}) (2 x^{2} + \frac{y^{2}}{400})$ using the FOIL Method.

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Step 1.8.1

Apply the distributive property.

$4 x^{4} - \frac{x^{2} y^{2}}{100} + \frac{y^{4}}{160000} - (2 x^{2} (2 x^{2} + \frac{y^{2}}{400}) + \frac{y^{2}}{400} (2 x^{2} + \frac{y^{2}}{400}))$

Step 1.8.2

Apply the distributive property.

$4 x^{4} - \frac{x^{2} y^{2}}{100} + \frac{y^{4}}{160000} - (2 x^{2} (2 x^{2}) + 2 x^{2} \frac{y^{2}}{400} + \frac{y^{2}}{400} (2 x^{2} + \frac{y^{2}}{400}))$

Step 1.8.3

Apply the distributive property.

$4 x^{4} - \frac{x^{2} y^{2}}{100} + \frac{y^{4}}{160000} - (2 x^{2} (2 x^{2}) + 2 x^{2} \frac{y^{2}}{400} + \frac{y^{2}}{400} (2 x^{2}) + \frac{y^{2}}{400} \cdot \frac{y^{2}}{400})$

Step 1.9

Simplify and combine like terms.

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Step 1.9.1

Simplify each term.

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Step 1.9.1.1

Rewrite using the commutative property of multiplication.

$4 x^{4} - \frac{x^{2} y^{2}}{100} + \frac{y^{4}}{160000} - (2 \cdot 2 x^{2} x^{2} + 2 x^{2} \frac{y^{2}}{400} + \frac{y^{2}}{400} (2 x^{2}) + \frac{y^{2}}{400} \cdot \frac{y^{2}}{400})$

Step 1.9.1.2

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

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Step 1.9.1.2.1

Move $x^{2}$ .

$4 x^{4} - \frac{x^{2} y^{2}}{100} + \frac{y^{4}}{160000} - (2 \cdot 2 (x^{2} x^{2}) + 2 x^{2} \frac{y^{2}}{400} + \frac{y^{2}}{400} (2 x^{2}) + \frac{y^{2}}{400} \cdot \frac{y^{2}}{400})$

Step 1.9.1.2.2

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

$4 x^{4} - \frac{x^{2} y^{2}}{100} + \frac{y^{4}}{160000} - (2 \cdot 2 x^{2 + 2} + 2 x^{2} \frac{y^{2}}{400} + \frac{y^{2}}{400} (2 x^{2}) + \frac{y^{2}}{400} \cdot \frac{y^{2}}{400})$

Step 1.9.1.2.3

Add $2$ and $2$ .

$4 x^{4} - \frac{x^{2} y^{2}}{100} + \frac{y^{4}}{160000} - (2 \cdot 2 x^{4} + 2 x^{2} \frac{y^{2}}{400} + \frac{y^{2}}{400} (2 x^{2}) + \frac{y^{2}}{400} \cdot \frac{y^{2}}{400})$

Step 1.9.1.3

Multiply $2$ by $2$ .

$4 x^{4} - \frac{x^{2} y^{2}}{100} + \frac{y^{4}}{160000} - (4 x^{4} + 2 x^{2} \frac{y^{2}}{400} + \frac{y^{2}}{400} (2 x^{2}) + \frac{y^{2}}{400} \cdot \frac{y^{2}}{400})$

Step 1.9.1.4

Cancel the common factor of $2$ .

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Step 1.9.1.4.1

Factor $2$ out of $2 x^{2}$ .

$4 x^{4} - \frac{x^{2} y^{2}}{100} + \frac{y^{4}}{160000} - (4 x^{4} + 2 (x^{2}) \frac{y^{2}}{400} + \frac{y^{2}}{400} (2 x^{2}) + \frac{y^{2}}{400} \cdot \frac{y^{2}}{400})$

Step 1.9.1.4.2

Factor $2$ out of $400$ .

$4 x^{4} - \frac{x^{2} y^{2}}{100} + \frac{y^{4}}{160000} - (4 x^{4} + 2 (x^{2}) \frac{y^{2}}{2 (200)} + \frac{y^{2}}{400} (2 x^{2}) + \frac{y^{2}}{400} \cdot \frac{y^{2}}{400})$

Step 1.9.1.4.3

Cancel the common factor.

$4 x^{4} - \frac{x^{2} y^{2}}{100} + \frac{y^{4}}{160000} - (4 x^{4} + 2 x^{2} \frac{y^{2}}{2 \cdot 200} + \frac{y^{2}}{400} (2 x^{2}) + \frac{y^{2}}{400} \cdot \frac{y^{2}}{400})$

Step 1.9.1.4.4

Rewrite the expression.

$4 x^{4} - \frac{x^{2} y^{2}}{100} + \frac{y^{4}}{160000} - (4 x^{4} + x^{2} \frac{y^{2}}{200} + \frac{y^{2}}{400} (2 x^{2}) + \frac{y^{2}}{400} \cdot \frac{y^{2}}{400})$

Step 1.9.1.5

Combine $x^{2}$ and $\frac{y^{2}}{200}$ .

$4 x^{4} - \frac{x^{2} y^{2}}{100} + \frac{y^{4}}{160000} - (4 x^{4} + \frac{x^{2} y^{2}}{200} + \frac{y^{2}}{400} (2 x^{2}) + \frac{y^{2}}{400} \cdot \frac{y^{2}}{400})$

Step 1.9.1.6

Rewrite using the commutative property of multiplication.

$4 x^{4} - \frac{x^{2} y^{2}}{100} + \frac{y^{4}}{160000} - (4 x^{4} + \frac{x^{2} y^{2}}{200} + 2 \frac{y^{2}}{400} x^{2} + \frac{y^{2}}{400} \cdot \frac{y^{2}}{400})$

Step 1.9.1.7

Cancel the common factor of $2$ .

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Step 1.9.1.7.1

Factor $2$ out of $400$ .

$4 x^{4} - \frac{x^{2} y^{2}}{100} + \frac{y^{4}}{160000} - (4 x^{4} + \frac{x^{2} y^{2}}{200} + 2 \frac{y^{2}}{2 (200)} x^{2} + \frac{y^{2}}{400} \cdot \frac{y^{2}}{400})$

Step 1.9.1.7.2

Cancel the common factor.

$4 x^{4} - \frac{x^{2} y^{2}}{100} + \frac{y^{4}}{160000} - (4 x^{4} + \frac{x^{2} y^{2}}{200} + 2 \frac{y^{2}}{2 \cdot 200} x^{2} + \frac{y^{2}}{400} \cdot \frac{y^{2}}{400})$

Step 1.9.1.7.3

Rewrite the expression.

$4 x^{4} - \frac{x^{2} y^{2}}{100} + \frac{y^{4}}{160000} - (4 x^{4} + \frac{x^{2} y^{2}}{200} + \frac{y^{2}}{200} x^{2} + \frac{y^{2}}{400} \cdot \frac{y^{2}}{400})$

Step 1.9.1.8

Combine $\frac{y^{2}}{200}$ and $x^{2}$ .

$4 x^{4} - \frac{x^{2} y^{2}}{100} + \frac{y^{4}}{160000} - (4 x^{4} + \frac{x^{2} y^{2}}{200} + \frac{y^{2} x^{2}}{200} + \frac{y^{2}}{400} \cdot \frac{y^{2}}{400})$

Step 1.9.1.9

Combine.

$4 x^{4} - \frac{x^{2} y^{2}}{100} + \frac{y^{4}}{160000} - (4 x^{4} + \frac{x^{2} y^{2}}{200} + \frac{y^{2} x^{2}}{200} + \frac{y^{2} y^{2}}{400 \cdot 400})$

Step 1.9.1.10

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

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Step 1.9.1.10.1

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

$4 x^{4} - \frac{x^{2} y^{2}}{100} + \frac{y^{4}}{160000} - (4 x^{4} + \frac{x^{2} y^{2}}{200} + \frac{y^{2} x^{2}}{200} + \frac{y^{2 + 2}}{400 \cdot 400})$

Step 1.9.1.10.2

Add $2$ and $2$ .

$4 x^{4} - \frac{x^{2} y^{2}}{100} + \frac{y^{4}}{160000} - (4 x^{4} + \frac{x^{2} y^{2}}{200} + \frac{y^{2} x^{2}}{200} + \frac{y^{4}}{400 \cdot 400})$

Step 1.9.1.11

Multiply $400$ by $400$ .

$4 x^{4} - \frac{x^{2} y^{2}}{100} + \frac{y^{4}}{160000} - (4 x^{4} + \frac{x^{2} y^{2}}{200} + \frac{y^{2} x^{2}}{200} + \frac{y^{4}}{160000})$

Step 1.9.2

Add $\frac{x^{2} y^{2}}{200}$ and $\frac{y^{2} x^{2}}{200}$ .

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Step 1.9.2.1

Reorder $y^{2}$ and $x^{2}$ .

$4 x^{4} - \frac{x^{2} y^{2}}{100} + \frac{y^{4}}{160000} - (4 x^{4} + \frac{x^{2} y^{2}}{200} + \frac{x^{2} y^{2}}{200} + \frac{y^{4}}{160000})$

Step 1.9.2.2

Add $\frac{x^{2} y^{2}}{200}$ and $\frac{x^{2} y^{2}}{200}$ .

$4 x^{4} - \frac{x^{2} y^{2}}{100} + \frac{y^{4}}{160000} - (4 x^{4} + 2 \frac{x^{2} y^{2}}{200} + \frac{y^{4}}{160000})$

Step 1.10

Cancel the common factor of $2$ .

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Step 1.10.1

Factor $2$ out of $200$ .

$4 x^{4} - \frac{x^{2} y^{2}}{100} + \frac{y^{4}}{160000} - (4 x^{4} + 2 \frac{x^{2} y^{2}}{2 (100)} + \frac{y^{4}}{160000})$

Step 1.10.2

Cancel the common factor.

$4 x^{4} - \frac{x^{2} y^{2}}{100} + \frac{y^{4}}{160000} - (4 x^{4} + 2 \frac{x^{2} y^{2}}{2 \cdot 100} + \frac{y^{4}}{160000})$

Step 1.10.3

Rewrite the expression.

$4 x^{4} - \frac{x^{2} y^{2}}{100} + \frac{y^{4}}{160000} - (4 x^{4} + \frac{x^{2} y^{2}}{100} + \frac{y^{4}}{160000})$

Step 1.11

Apply the distributive property.

$4 x^{4} - \frac{x^{2} y^{2}}{100} + \frac{y^{4}}{160000} - (4 x^{4}) - \frac{x^{2} y^{2}}{100} - \frac{y^{4}}{160000}$

Step 1.12

Multiply $4$ by $- 1$ .

$4 x^{4} - \frac{x^{2} y^{2}}{100} + \frac{y^{4}}{160000} - 4 x^{4} - \frac{x^{2} y^{2}}{100} - \frac{y^{4}}{160000}$

Step 2

Simplify terms.

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Step 2.1

Combine the opposite terms in $4 x^{4} - \frac{x^{2} y^{2}}{100} + \frac{y^{4}}{160000} - 4 x^{4} - \frac{x^{2} y^{2}}{100} - \frac{y^{4}}{160000}$ .

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Step 2.1.1

Subtract $4 x^{4}$ from $4 x^{4}$ .

$- \frac{x^{2} y^{2}}{100} + \frac{y^{4}}{160000} + 0 - \frac{x^{2} y^{2}}{100} - \frac{y^{4}}{160000}$

Step 2.1.2

Add $- \frac{x^{2} y^{2}}{100} + \frac{y^{4}}{160000}$ and $0$ .

$- \frac{x^{2} y^{2}}{100} + \frac{y^{4}}{160000} - \frac{x^{2} y^{2}}{100} - \frac{y^{4}}{160000}$

Step 2.1.3

Subtract $\frac{y^{4}}{160000}$ from $\frac{y^{4}}{160000}$ .

$- \frac{x^{2} y^{2}}{100} - \frac{x^{2} y^{2}}{100} + 0$

Step 2.1.4

Add $- \frac{x^{2} y^{2}}{100} - \frac{x^{2} y^{2}}{100}$ and $0$ .

$- \frac{x^{2} y^{2}}{100} - \frac{x^{2} y^{2}}{100}$

Step 2.2

Combine the numerators over the common denominator.

$\frac{- x^{2} y^{2} - x^{2} y^{2}}{100}$

Step 2.3

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

$\frac{- 2 x^{2} y^{2}}{100}$

Step 2.4

Cancel the common factor of $- 2$ and $100$ .

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Step 2.4.1

Factor $2$ out of $- 2 x^{2} y^{2}$ .

$\frac{2 (- x^{2} y^{2})}{100}$

Step 2.4.2

Cancel the common factors.

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Step 2.4.2.1

Factor $2$ out of $100$ .

$\frac{2 (- x^{2} y^{2})}{2 (50)}$

Step 2.4.2.2

Cancel the common factor.

$\frac{2 (- x^{2} y^{2})}{2 \cdot 50}$

Step 2.4.2.3

Rewrite the expression.

$\frac{- x^{2} y^{2}}{50}$

Step 2.5

Move the negative in front of the fraction.

$- \frac{x^{2} y^{2}}{50}$