Basic Math 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

Simplify terms.

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Step 3.1

Cancel the common factor of $u - v$ .

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Step 3.1.1

Cancel the common factor.

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

Step 3.1.2

Divide $u + v$ by $1$ .

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

Factor $u$ out of $- v u$ .

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

Step 3.2.3

Factor $u$ out of $u \cdot u + u (- v)$ .

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

Step 3.3

Cancel the common factor of $u$ .

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Step 3.3.1

Cancel the common factor.

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

Step 3.3.2

Divide $u - v$ by $1$ .

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

Step 4

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

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Step 4.1

Apply the distributive property.

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Apply the distributive property.

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

Step 5

Simplify terms.

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Step 5.1

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

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Step 5.1.1

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

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

Step 5.1.2

Add $- u v$ and $u v$ .

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

Step 5.1.3

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

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

Step 5.2

Simplify each term.

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Step 5.2.1

Multiply $u$ by $u$ .

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

Step 5.2.2

Rewrite using the commutative property of multiplication.

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

Step 5.2.3

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

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Step 5.2.3.1

Move $v$ .

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

Step 5.2.3.2

Multiply $v$ by $v$ .

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