Pre-Algebra Examples

$\frac{w - \frac{1}{w}}{w \div \frac{1}{w}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$(w - \frac{1}{w}) \frac{\frac{1}{w}}{w}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

$(w - \frac{1}{w}) (\frac{1}{w} \cdot \frac{1}{w})$

Step 3

Multiply $\frac{1}{w} \cdot \frac{1}{w}$ .

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Step 3.1

Multiply $\frac{1}{w}$ by $\frac{1}{w}$ .

$(w - \frac{1}{w}) \frac{1}{w \cdot w}$

Step 3.2

Raise $w$ to the power of $1$ .

$(w - \frac{1}{w}) \frac{1}{w^{1} w}$

Step 3.3

Raise $w$ to the power of $1$ .

$(w - \frac{1}{w}) \frac{1}{w^{1} w^{1}}$

Step 3.4

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

$(w - \frac{1}{w}) \frac{1}{w^{1 + 1}}$

Step 3.5

Add $1$ and $1$ .

$(w - \frac{1}{w}) \frac{1}{w^{2}}$

Step 4

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

$\frac{w - \frac{1}{w}}{w^{2}}$

Step 5

Simplify the numerator.

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Step 5.1

To write $w$ as a fraction with a common denominator, multiply by $\frac{w}{w}$ .

$\frac{\frac{w \cdot w}{w} - \frac{1}{w}}{w^{2}}$

Step 5.2

Combine the numerators over the common denominator.

$\frac{\frac{w \cdot w - 1}{w}}{w^{2}}$

Step 5.3

Simplify the numerator.

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Step 5.3.1

Raise $w$ to the power of $1$ .

$\frac{\frac{w^{1} w - 1}{w}}{w^{2}}$

Step 5.3.2

Raise $w$ to the power of $1$ .

$\frac{\frac{w^{1} w^{1} - 1}{w}}{w^{2}}$

Step 5.3.3

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

$\frac{\frac{w^{1 + 1} - 1}{w}}{w^{2}}$

Step 5.3.4

Add $1$ and $1$ .

$\frac{\frac{w^{2} - 1}{w}}{w^{2}}$

Step 5.3.5

Rewrite $1$ as $1^{2}$ .

$\frac{\frac{w^{2} - 1^{2}}{w}}{w^{2}}$

Step 5.3.6

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = w$ and $b = 1$ .

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

Step 6

Multiply the numerator by the reciprocal of the denominator.

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

Step 7

Combine.

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

Step 8

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

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Step 8.1

Multiply $w$ by $w^{2}$ .

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Step 8.1.1

Raise $w$ to the power of $1$ .

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

Step 8.1.2

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

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

Step 8.2

Add $1$ and $2$ .

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

Step 9

Multiply $(w + 1) (w - 1)$ by $1$ .

$\frac{(w + 1) (w - 1)}{w^{3}}$

Step 10

Expand $(w + 1) (w - 1)$ using the FOIL Method.

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Step 10.1

Apply the distributive property.

$\frac{w (w - 1) + 1 (w - 1)}{w^{3}}$

Step 10.2

Apply the distributive property.

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

Step 10.3

Apply the distributive property.

$\frac{w \cdot w + w \cdot - 1 + 1 w + 1 \cdot - 1}{w^{3}}$

Step 11

Simplify and combine like terms.

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Step 11.1

Simplify each term.

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Step 11.1.1

Multiply $w$ by $w$ .

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

Step 11.1.2

Move $- 1$ to the left of $w$ .

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

Step 11.1.3

Rewrite $- 1 w$ as $- w$ .

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

Step 11.1.4

Multiply $w$ by $1$ .

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

Step 11.1.5

Multiply $- 1$ by $1$ .

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

Step 11.2

Add $- w$ and $w$ .

$\frac{w^{2} + 0 - 1}{w^{3}}$

Step 11.3

Add $w^{2}$ and $0$ .

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

Step 12

Split the fraction $\frac{w^{2} - 1}{w^{3}}$ into two fractions.

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

Step 13

Cancel the common factor of $w^{2}$ and $w^{3}$ .

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Step 13.1

Multiply by $1$ .

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

Step 13.2

Cancel the common factors.

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Step 13.2.1

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

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

Step 13.2.2

Cancel the common factor.

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

Step 13.2.3

Rewrite the expression.

$\frac{1}{w} + \frac{- 1}{w^{3}}$

Step 14

Move the negative in front of the fraction.

$\frac{1}{w} - \frac{1}{w^{3}}$