Calculus Examples

$(v^{3} - 2 v) (v^{- 4} + v^{- 2})$

Step 1

Simplify each term.

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Step 1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 1.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2

Expand $(v^{3} - 2 v) (\frac{1}{v^{4}} + \frac{1}{v^{2}})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

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

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

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Step 3.1.1.1

Factor $v^{3}$ out of $v^{4}$ .

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

Step 3.1.1.2

Cancel the common factor.

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

Step 3.1.1.3

Rewrite the expression.

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

Step 3.1.2

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

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Step 3.1.2.1

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

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

Step 3.1.2.2

Cancel the common factor.

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

Step 3.1.2.3

Rewrite the expression.

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

Step 3.1.3

Cancel the common factor of $v$ .

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Step 3.1.3.1

Factor $v$ out of $- 2 v$ .

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

Step 3.1.3.2

Factor $v$ out of $v^{4}$ .

$\frac{1}{v} + v + v \cdot - 2 \frac{1}{v \cdot v^{3}} - 2 v \frac{1}{v^{2}}$

Step 3.1.3.3

Cancel the common factor.

$\frac{1}{v} + v + v \cdot - 2 \frac{1}{v \cdot v^{3}} - 2 v \frac{1}{v^{2}}$

Step 3.1.3.4

Rewrite the expression.

$\frac{1}{v} + v - 2 \frac{1}{v^{3}} - 2 v \frac{1}{v^{2}}$

Step 3.1.4

Combine $- 2$ and $\frac{1}{v^{3}}$ .

$\frac{1}{v} + v + \frac{- 2}{v^{3}} - 2 v \frac{1}{v^{2}}$

Step 3.1.5

Move the negative in front of the fraction.

$\frac{1}{v} + v - \frac{2}{v^{3}} - 2 v \frac{1}{v^{2}}$

Step 3.1.6

Cancel the common factor of $v$ .

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Step 3.1.6.1

Factor $v$ out of $- 2 v$ .

$\frac{1}{v} + v - \frac{2}{v^{3}} + v \cdot - 2 \frac{1}{v^{2}}$

Step 3.1.6.2

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

$\frac{1}{v} + v - \frac{2}{v^{3}} + v \cdot - 2 \frac{1}{v \cdot v}$

Step 3.1.6.3

Cancel the common factor.

$\frac{1}{v} + v - \frac{2}{v^{3}} + v \cdot - 2 \frac{1}{v \cdot v}$

Step 3.1.6.4

Rewrite the expression.

$\frac{1}{v} + v - \frac{2}{v^{3}} - 2 \frac{1}{v}$

Step 3.1.7

Combine $- 2$ and $\frac{1}{v}$ .

$\frac{1}{v} + v - \frac{2}{v^{3}} + \frac{- 2}{v}$

Step 3.1.8

Move the negative in front of the fraction.

$\frac{1}{v} + v - \frac{2}{v^{3}} - \frac{2}{v}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{1 - 2}{v} + v - \frac{2}{v^{3}}$

Step 3.3

Subtract $2$ from $1$ .

$\frac{- 1}{v} + v - \frac{2}{v^{3}}$

Step 4

Move the negative in front of the fraction.

$- \frac{1}{v} + v - \frac{2}{v^{3}}$

Step 5

Reorder $- \frac{1}{v}$ and $v$ .

$v - \frac{1}{v} - \frac{2}{v^{3}}$