Basic Math Examples

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

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

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}

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

Step 2

Since both terms are perfect squares, factor using the difference of squares formula,

a^{2} - b^{2} = (a + b) (a - b)

$a^{2} - b^{2} = (a + b) (a - b)$ where

a = u

$a = u$ and

b = v

$b = v$ .

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

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

Step 3

Simplify terms.

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Step 3.1

Cancel the common factor of

u - v

$u - v$ .

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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

$v$ .

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