Calculus Examples

${(\frac{y^{3}}{2} - \frac{1}{2 y^{3}})}^{2}$

Step 1

Rewrite ${(\frac{y^{3}}{2} - \frac{1}{2 y^{3}})}^{2}$ as $(\frac{y^{3}}{2} - \frac{1}{2 y^{3}}) (\frac{y^{3}}{2} - \frac{1}{2 y^{3}})$ .

$(\frac{y^{3}}{2} - \frac{1}{2 y^{3}}) (\frac{y^{3}}{2} - \frac{1}{2 y^{3}})$

Step 2

Expand $(\frac{y^{3}}{2} - \frac{1}{2 y^{3}}) (\frac{y^{3}}{2} - \frac{1}{2 y^{3}})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$\frac{y^{3}}{2} (\frac{y^{3}}{2} - \frac{1}{2 y^{3}}) - \frac{1}{2 y^{3}} (\frac{y^{3}}{2} - \frac{1}{2 y^{3}})$

Step 2.2

Apply the distributive property.

$\frac{y^{3}}{2} \cdot \frac{y^{3}}{2} + \frac{y^{3}}{2} (- \frac{1}{2 y^{3}}) - \frac{1}{2 y^{3}} (\frac{y^{3}}{2} - \frac{1}{2 y^{3}})$

Step 2.3

Apply the distributive property.

$\frac{y^{3}}{2} \cdot \frac{y^{3}}{2} + \frac{y^{3}}{2} (- \frac{1}{2 y^{3}}) - \frac{1}{2 y^{3}} \cdot \frac{y^{3}}{2} - \frac{1}{2 y^{3}} (- \frac{1}{2 y^{3}})$

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.

$\frac{y^{3} y^{3}}{2 \cdot 2} + \frac{y^{3}}{2} (- \frac{1}{2 y^{3}}) - \frac{1}{2 y^{3}} \cdot \frac{y^{3}}{2} - \frac{1}{2 y^{3}} (- \frac{1}{2 y^{3}})$

Step 3.1.2

Multiply $y^{3}$ by $y^{3}$ 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.

$\frac{y^{3 + 3}}{2 \cdot 2} + \frac{y^{3}}{2} (- \frac{1}{2 y^{3}}) - \frac{1}{2 y^{3}} \cdot \frac{y^{3}}{2} - \frac{1}{2 y^{3}} (- \frac{1}{2 y^{3}})$

Step 3.1.2.2

Add $3$ and $3$ .

$\frac{y^{6}}{2 \cdot 2} + \frac{y^{3}}{2} (- \frac{1}{2 y^{3}}) - \frac{1}{2 y^{3}} \cdot \frac{y^{3}}{2} - \frac{1}{2 y^{3}} (- \frac{1}{2 y^{3}})$

Step 3.1.3

Multiply $2$ by $2$ .

$\frac{y^{6}}{4} + \frac{y^{3}}{2} (- \frac{1}{2 y^{3}}) - \frac{1}{2 y^{3}} \cdot \frac{y^{3}}{2} - \frac{1}{2 y^{3}} (- \frac{1}{2 y^{3}})$

Step 3.1.4

Rewrite using the commutative property of multiplication.

$\frac{y^{6}}{4} - \frac{y^{3}}{2} \cdot \frac{1}{2 y^{3}} - \frac{1}{2 y^{3}} \cdot \frac{y^{3}}{2} - \frac{1}{2 y^{3}} (- \frac{1}{2 y^{3}})$

Step 3.1.5

Cancel the common factor of $y^{3}$ .

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Step 3.1.5.1

Move the leading negative in $- \frac{y^{3}}{2}$ into the numerator.

$\frac{y^{6}}{4} + \frac{- y^{3}}{2} \cdot \frac{1}{2 y^{3}} - \frac{1}{2 y^{3}} \cdot \frac{y^{3}}{2} - \frac{1}{2 y^{3}} (- \frac{1}{2 y^{3}})$

Step 3.1.5.2

Factor $y^{3}$ out of $- y^{3}$ .

$\frac{y^{6}}{4} + \frac{y^{3} \cdot - 1}{2} \cdot \frac{1}{2 y^{3}} - \frac{1}{2 y^{3}} \cdot \frac{y^{3}}{2} - \frac{1}{2 y^{3}} (- \frac{1}{2 y^{3}})$

Step 3.1.5.3

Factor $y^{3}$ out of $2 y^{3}$ .

$\frac{y^{6}}{4} + \frac{y^{3} \cdot - 1}{2} \cdot \frac{1}{y^{3} \cdot 2} - \frac{1}{2 y^{3}} \cdot \frac{y^{3}}{2} - \frac{1}{2 y^{3}} (- \frac{1}{2 y^{3}})$

Step 3.1.5.4

Cancel the common factor.

$\frac{y^{6}}{4} + \frac{y^{3} \cdot - 1}{2} \cdot \frac{1}{y^{3} \cdot 2} - \frac{1}{2 y^{3}} \cdot \frac{y^{3}}{2} - \frac{1}{2 y^{3}} (- \frac{1}{2 y^{3}})$

Step 3.1.5.5

Rewrite the expression.

$\frac{y^{6}}{4} + \frac{- 1}{2} \cdot \frac{1}{2} - \frac{1}{2 y^{3}} \cdot \frac{y^{3}}{2} - \frac{1}{2 y^{3}} (- \frac{1}{2 y^{3}})$

Step 3.1.6

Multiply $\frac{- 1}{2}$ by $\frac{1}{2}$ .

$\frac{y^{6}}{4} + \frac{- 1}{2 \cdot 2} - \frac{1}{2 y^{3}} \cdot \frac{y^{3}}{2} - \frac{1}{2 y^{3}} (- \frac{1}{2 y^{3}})$

Step 3.1.7

Multiply $2$ by $2$ .

$\frac{y^{6}}{4} + \frac{- 1}{4} - \frac{1}{2 y^{3}} \cdot \frac{y^{3}}{2} - \frac{1}{2 y^{3}} (- \frac{1}{2 y^{3}})$

Step 3.1.8

Move the negative in front of the fraction.

$\frac{y^{6}}{4} - \frac{1}{4} - \frac{1}{2 y^{3}} \cdot \frac{y^{3}}{2} - \frac{1}{2 y^{3}} (- \frac{1}{2 y^{3}})$

Step 3.1.9

Cancel the common factor of $y^{3}$ .

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Step 3.1.9.1

Move the leading negative in $- \frac{1}{2 y^{3}}$ into the numerator.

$\frac{y^{6}}{4} - \frac{1}{4} + \frac{- 1}{2 y^{3}} \cdot \frac{y^{3}}{2} - \frac{1}{2 y^{3}} (- \frac{1}{2 y^{3}})$

Step 3.1.9.2

Factor $y^{3}$ out of $2 y^{3}$ .

$\frac{y^{6}}{4} - \frac{1}{4} + \frac{- 1}{y^{3} \cdot 2} \cdot \frac{y^{3}}{2} - \frac{1}{2 y^{3}} (- \frac{1}{2 y^{3}})$

Step 3.1.9.3

Cancel the common factor.

$\frac{y^{6}}{4} - \frac{1}{4} + \frac{- 1}{y^{3} \cdot 2} \cdot \frac{y^{3}}{2} - \frac{1}{2 y^{3}} (- \frac{1}{2 y^{3}})$

Step 3.1.9.4

Rewrite the expression.

$\frac{y^{6}}{4} - \frac{1}{4} + \frac{- 1}{2} \cdot \frac{1}{2} - \frac{1}{2 y^{3}} (- \frac{1}{2 y^{3}})$

Step 3.1.10

Multiply $\frac{- 1}{2}$ by $\frac{1}{2}$ .

$\frac{y^{6}}{4} - \frac{1}{4} + \frac{- 1}{2 \cdot 2} - \frac{1}{2 y^{3}} (- \frac{1}{2 y^{3}})$

Step 3.1.11

Multiply $2$ by $2$ .

$\frac{y^{6}}{4} - \frac{1}{4} + \frac{- 1}{4} - \frac{1}{2 y^{3}} (- \frac{1}{2 y^{3}})$

Step 3.1.12

Move the negative in front of the fraction.

$\frac{y^{6}}{4} - \frac{1}{4} - \frac{1}{4} - \frac{1}{2 y^{3}} (- \frac{1}{2 y^{3}})$

Step 3.1.13

Multiply $- \frac{1}{2 y^{3}} (- \frac{1}{2 y^{3}})$ .

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Step 3.1.13.1

Multiply $- 1$ by $- 1$ .

$\frac{y^{6}}{4} - \frac{1}{4} - \frac{1}{4} + 1 \frac{1}{2 y^{3}} \frac{1}{2 y^{3}}$

Step 3.1.13.2

Multiply $\frac{1}{2 y^{3}}$ by $1$ .

$\frac{y^{6}}{4} - \frac{1}{4} - \frac{1}{4} + \frac{1}{2 y^{3}} \cdot \frac{1}{2 y^{3}}$

Step 3.1.13.3

Multiply $\frac{1}{2 y^{3}}$ by $\frac{1}{2 y^{3}}$ .

$\frac{y^{6}}{4} - \frac{1}{4} - \frac{1}{4} + \frac{1}{2 y^{3} (2 y^{3})}$

Step 3.1.13.4

Multiply $2$ by $2$ .

$\frac{y^{6}}{4} - \frac{1}{4} - \frac{1}{4} + \frac{1}{4 y^{3} y^{3}}$

Step 3.1.13.5

Multiply $y^{3}$ by $y^{3}$ by adding the exponents.

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Step 3.1.13.5.1

Move $y^{3}$ .

$\frac{y^{6}}{4} - \frac{1}{4} - \frac{1}{4} + \frac{1}{4 (y^{3} y^{3})}$

Step 3.1.13.5.2

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

$\frac{y^{6}}{4} - \frac{1}{4} - \frac{1}{4} + \frac{1}{4 y^{3 + 3}}$

Step 3.1.13.5.3

Add $3$ and $3$ .

$\frac{y^{6}}{4} - \frac{1}{4} - \frac{1}{4} + \frac{1}{4 y^{6}}$

Step 3.2

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

$\frac{y^{6}}{4} - 2 (\frac{1}{4}) + \frac{1}{4 y^{6}}$

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 $- 2$ .

$\frac{y^{6}}{4} + 2 (- 1) \frac{1}{4} + \frac{1}{4 y^{6}}$

Step 4.1.2

Factor $2$ out of $4$ .

$\frac{y^{6}}{4} + 2 \cdot - 1 \frac{1}{2 \cdot 2} + \frac{1}{4 y^{6}}$

Step 4.1.3

Cancel the common factor.

$\frac{y^{6}}{4} + 2 \cdot - 1 \frac{1}{2 \cdot 2} + \frac{1}{4 y^{6}}$

Step 4.1.4

Rewrite the expression.

$\frac{y^{6}}{4} - 1 (\frac{1}{2}) + \frac{1}{4 y^{6}}$

Step 4.2

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

$\frac{y^{6}}{4} - \frac{1}{2} + \frac{1}{4 y^{6}}$