Basic Math Examples

$\frac{\frac{1}{u^{2}} - \frac{1}{v^{2}}}{\frac{7}{u} + \frac{7}{v}}$

Step 1

Multiply the numerator and denominator of the fraction by $u^{2} v^{2}$ .

Tap for more steps...

Step 1.1

Multiply $\frac{\frac{1}{u^{2}} - \frac{1}{v^{2}}}{\frac{7}{u} + \frac{7}{v}}$ by $\frac{u^{2} v^{2}}{u^{2} v^{2}}$ .

$\frac{u^{2} v^{2}}{u^{2} v^{2}} \cdot \frac{\frac{1}{u^{2}} - \frac{1}{v^{2}}}{\frac{7}{u} + \frac{7}{v}}$

Step 1.2

Combine.

$\frac{u^{2} v^{2} (\frac{1}{u^{2}} - \frac{1}{v^{2}})}{u^{2} v^{2} (\frac{7}{u} + \frac{7}{v})}$

Step 2

Apply the distributive property.

$\frac{u^{2} v^{2} \frac{1}{u^{2}} + u^{2} v^{2} (- \frac{1}{v^{2}})}{u^{2} v^{2} \frac{7}{u} + u^{2} v^{2} \frac{7}{v}}$

Step 3

Simplify by cancelling.

Tap for more steps...

Step 3.1

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

Tap for more steps...

Step 3.1.1

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

$\frac{u^{2} (v^{2}) \frac{1}{u^{2}} + u^{2} v^{2} (- \frac{1}{v^{2}})}{u^{2} v^{2} \frac{7}{u} + u^{2} v^{2} \frac{7}{v}}$

Step 3.1.2

Cancel the common factor.

$\frac{u^{2} v^{2} \frac{1}{u^{2}} + u^{2} v^{2} (- \frac{1}{v^{2}})}{u^{2} v^{2} \frac{7}{u} + u^{2} v^{2} \frac{7}{v}}$

Step 3.1.3

Rewrite the expression.

$\frac{v^{2} + u^{2} v^{2} (- \frac{1}{v^{2}})}{u^{2} v^{2} \frac{7}{u} + u^{2} v^{2} \frac{7}{v}}$

Step 3.2

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

Tap for more steps...

Step 3.2.1

Move the leading negative in $- \frac{1}{v^{2}}$ into the numerator.

$\frac{v^{2} + u^{2} v^{2} \frac{- 1}{v^{2}}}{u^{2} v^{2} \frac{7}{u} + u^{2} v^{2} \frac{7}{v}}$

Step 3.2.2

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

$\frac{v^{2} + v^{2} u^{2} \frac{- 1}{v^{2}}}{u^{2} v^{2} \frac{7}{u} + u^{2} v^{2} \frac{7}{v}}$

Step 3.2.3

Cancel the common factor.

$\frac{v^{2} + v^{2} u^{2} \frac{- 1}{v^{2}}}{u^{2} v^{2} \frac{7}{u} + u^{2} v^{2} \frac{7}{v}}$

Step 3.2.4

Rewrite the expression.

$\frac{v^{2} + u^{2} \cdot - 1}{u^{2} v^{2} \frac{7}{u} + u^{2} v^{2} \frac{7}{v}}$

Step 3.3

Cancel the common factor of $u$ .

Tap for more steps...

Step 3.3.1

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

$\frac{v^{2} + u^{2} \cdot - 1}{u (u v^{2}) \frac{7}{u} + u^{2} v^{2} \frac{7}{v}}$

Step 3.3.2

Cancel the common factor.

$\frac{v^{2} + u^{2} \cdot - 1}{u (u v^{2}) \frac{7}{u} + u^{2} v^{2} \frac{7}{v}}$

Step 3.3.3

Rewrite the expression.

$\frac{v^{2} + u^{2} \cdot - 1}{u v^{2} \cdot 7 + u^{2} v^{2} \frac{7}{v}}$

Step 3.4

Cancel the common factor of $v$ .

Tap for more steps...

Step 3.4.1

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

$\frac{v^{2} + u^{2} \cdot - 1}{u v^{2} \cdot 7 + v (u^{2} v) \frac{7}{v}}$

Step 3.4.2

Cancel the common factor.

$\frac{v^{2} + u^{2} \cdot - 1}{u v^{2} \cdot 7 + v (u^{2} v) \frac{7}{v}}$

Step 3.4.3

Rewrite the expression.

$\frac{v^{2} + u^{2} \cdot - 1}{u v^{2} \cdot 7 + u^{2} v \cdot 7}$

Step 4

Simplify the numerator.

Tap for more steps...

Step 4.1

Move $- 1$ to the left of $u^{2}$ .

$\frac{v^{2} - 1 \cdot u^{2}}{u v^{2} \cdot 7 + u^{2} v \cdot 7}$

Step 4.2

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

$\frac{v^{2} - u^{2}}{u v^{2} \cdot 7 + u^{2} v \cdot 7}$

Step 4.3

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

$\frac{(v + u) (v - u)}{u v^{2} \cdot 7 + u^{2} v \cdot 7}$

Step 5

Simplify terms.

Tap for more steps...

Step 5.1

Factor $u v \cdot 7$ out of $u v^{2} \cdot 7 + u^{2} v \cdot 7$ .

Tap for more steps...

Step 5.1.1

Factor $u v \cdot 7$ out of $u v^{2} \cdot 7$ .

$\frac{(v + u) (v - u)}{u v \cdot 7 (v) + u^{2} v \cdot 7}$

Step 5.1.2

Factor $u v \cdot 7$ out of $u^{2} v \cdot 7$ .

$\frac{(v + u) (v - u)}{u v \cdot 7 (v) + u v \cdot 7 (u)}$

Step 5.1.3

Factor $u v \cdot 7$ out of $u v \cdot 7 (v) + u v \cdot 7 (u)$ .

$\frac{(v + u) (v - u)}{u v \cdot 7 (v + u)}$

Step 5.2

Cancel the common factor of $v + u$ .

Tap for more steps...

Step 5.2.1

Cancel the common factor.

$\frac{(v + u) (v - u)}{u v \cdot 7 (v + u)}$

Step 5.2.2

Rewrite the expression.

$\frac{v - u}{u v \cdot 7}$

Step 5.3

Move $7$ to the left of $u v$ .

$\frac{v - u}{7 u v}$