Calculus Examples

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

Step 1

Rewrite ${(x^{2} - \frac{1}{4 x^{2}})}^{2}$ as $(x^{2} - \frac{1}{4 x^{2}}) (x^{2} - \frac{1}{4 x^{2}})$ .

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

Step 2

Expand $(x^{2} - \frac{1}{4 x^{2}}) (x^{2} - \frac{1}{4 x^{2}})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

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

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Step 3.1.1.1

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

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

Step 3.1.1.2

Add $2$ and $2$ .

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

Step 3.1.2

Rewrite using the commutative property of multiplication.

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

Step 3.1.3

Cancel the common factor of $x^{2}$ .

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Step 3.1.3.1

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

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

Step 3.1.3.2

Factor $x^{2}$ out of $4 x^{2}$ .

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

Step 3.1.3.3

Cancel the common factor.

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

Step 3.1.3.4

Rewrite the expression.

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

Step 3.1.4

Cancel the common factor of $x^{2}$ .

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Step 3.1.4.1

Move the leading negative in $- \frac{1}{4 x^{2}}$ into the numerator.

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

Step 3.1.4.2

Factor $x^{2}$ out of $4 x^{2}$ .

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

Step 3.1.4.3

Cancel the common factor.

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

Step 3.1.4.4

Rewrite the expression.

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

Step 3.1.5

Move the negative in front of the fraction.

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

Step 3.1.6

Multiply $- \frac{1}{4 x^{2}} (- \frac{1}{4 x^{2}})$ .

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Step 3.1.6.1

Multiply $- 1$ by $- 1$ .

$x^{4} - \frac{1}{4} - \frac{1}{4} + 1 \frac{1}{4 x^{2}} \frac{1}{4 x^{2}}$

Step 3.1.6.2

Multiply $\frac{1}{4 x^{2}}$ by $1$ .

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

Step 3.1.6.3

Multiply $\frac{1}{4 x^{2}}$ by $\frac{1}{4 x^{2}}$ .

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

Step 3.1.6.4

Multiply $4$ by $4$ .

$x^{4} - \frac{1}{4} - \frac{1}{4} + \frac{1}{16 x^{2} x^{2}}$

Step 3.1.6.5

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

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Step 3.1.6.5.1

Move $x^{2}$ .

$x^{4} - \frac{1}{4} - \frac{1}{4} + \frac{1}{16 (x^{2} x^{2})}$

Step 3.1.6.5.2

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

$x^{4} - \frac{1}{4} - \frac{1}{4} + \frac{1}{16 x^{2 + 2}}$

Step 3.1.6.5.3

Add $2$ and $2$ .

$x^{4} - \frac{1}{4} - \frac{1}{4} + \frac{1}{16 x^{4}}$

Step 3.2

Subtract $\frac{1}{4}$ from $- \frac{1}{4}$ .

$x^{4} - 2 (\frac{1}{4}) + \frac{1}{16 x^{4}}$

Step 4

Simplify each term.

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Step 4.1

Cancel the common factor of $2$ .

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Step 4.1.1

Factor $2$ out of $- 2$ .

$x^{4} + 2 (- 1) \frac{1}{4} + \frac{1}{16 x^{4}}$

Step 4.1.2

Factor $2$ out of $4$ .

$x^{4} + 2 \cdot - 1 \frac{1}{2 \cdot 2} + \frac{1}{16 x^{4}}$

Step 4.1.3

Cancel the common factor.

$x^{4} + 2 \cdot - 1 \frac{1}{2 \cdot 2} + \frac{1}{16 x^{4}}$

Step 4.1.4

Rewrite the expression.

$x^{4} - 1 (\frac{1}{2}) + \frac{1}{16 x^{4}}$

Step 4.2

Rewrite $- 1 (\frac{1}{2})$ as $- (\frac{1}{2})$ .

$x^{4} - \frac{1}{2} + \frac{1}{16 x^{4}}$