Pre-Algebra Examples

$\frac{u^{2} - v^{2}}{u + v} \div \frac{u}{u^{2} + v u}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{u^{2} - v^{2}}{u + v} \cdot \frac{u^{2} + v u}{u}$

Step 2

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

$\frac{(u + v) (u - v)}{u + v} \cdot \frac{u^{2} + v u}{u}$

Step 3

Cancel the common factor of $u + v$ .

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Step 3.1

Cancel the common factor.

$\frac{(u + v) (u - v)}{u + v} \cdot \frac{u^{2} + v u}{u}$

Step 3.2

Divide $u - v$ by $1$ .

$(u - v) \frac{u^{2} + v u}{u}$

Step 4

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

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Step 4.1

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

$(u - v) \frac{u \cdot u + v u}{u}$

Step 4.2

Factor $u$ out of $v u$ .

$(u - v) \frac{u \cdot u + u v}{u}$

Step 4.3

Factor $u$ out of $u \cdot u + u v$ .

$(u - v) \frac{u (u + v)}{u}$

Step 5

Cancel the common factor of $u$ .

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Step 5.1

Cancel the common factor.

$(u - v) \frac{u (u + v)}{u}$

Step 5.2

Divide $u + v$ by $1$ .

$(u - v) (u + v)$

Step 6

Expand $(u - v) (u + v)$ using the FOIL Method.

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Step 6.1

Apply the distributive property.

$u (u + v) - v (u + v)$

Step 6.2

Apply the distributive property.

$u \cdot u + u v - v (u + v)$

Step 6.3

Apply the distributive property.

$u \cdot u + u v - v u - v \cdot v$

Step 7

Combine the opposite terms in $u \cdot u + u v - v u - v \cdot v$ .

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Step 7.1

Reorder the factors in the terms $u v$ and $- v u$ .

$u \cdot u + u v - u v - v \cdot v$

Step 7.2

Subtract $u v$ from $u v$ .

$u \cdot u + 0 - v \cdot v$

Step 7.3

Add $u \cdot u$ and $0$ .

$u \cdot u - v \cdot v$

Step 8

Simplify each term.

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Step 8.1

Multiply $u$ by $u$ .

$u^{2} - v \cdot v$

Step 8.2

Multiply $v$ by $v$ by adding the exponents.

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Step 8.2.1

Move $v$ .

$u^{2} - (v \cdot v)$

Step 8.2.2

Multiply $v$ by $v$ .

$u^{2} - v^{2}$