Algebra Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

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

Combine.

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

Step 3.1.2

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

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Step 3.1.2.1

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

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

Step 3.1.2.2

Add $2$ and $2$ .

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

Step 3.1.3

Multiply $4$ by $4$ .

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

Step 3.1.4

Rewrite using the commutative property of multiplication.

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

Step 3.1.5

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

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Step 3.1.5.1

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

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

Step 3.1.5.2

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

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

Step 3.1.5.3

Cancel the common factor.

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

Step 3.1.5.4

Rewrite the expression.

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

Step 3.1.6

Move the negative in front of the fraction.

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

Step 3.1.7

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

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Step 3.1.7.1

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

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

Step 3.1.7.2

Cancel the common factor.

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

Step 3.1.7.3

Rewrite the expression.

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

Step 3.1.8

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

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

Step 3.1.9

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

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Step 3.1.9.1

Multiply $- 1$ by $- 1$ .

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

Step 3.1.9.2

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

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

Step 3.1.9.3

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

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

Step 3.1.9.4

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

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Step 3.1.9.4.1

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

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

Step 3.1.9.4.2

Add $2$ and $2$ .

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

Step 3.2

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

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

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 $- 2$ .

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

Step 4.1.2

Factor $2$ out of $4$ .

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

Step 4.1.3

Cancel the common factor.

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

Step 4.1.4

Rewrite the expression.

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

Step 4.2

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

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