Basic Math Examples

$\frac{u w - u t + 3 v w - 3 v t}{w^{2} - t^{2}} \cdot \frac{6 w + 6 t + u w + t u}{36 - u^{2}}$

Step 1

Simplify the numerator.

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Step 1.1

Factor out the greatest common factor from each group.

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Step 1.1.1

Group the first two terms and the last two terms.

$\frac{(u w - u t) + 3 v w - 3 v t}{w^{2} - t^{2}} \cdot \frac{6 w + 6 t + u w + t u}{36 - u^{2}}$

Step 1.1.2

Factor out the greatest common factor (GCF) from each group.

$\frac{u (w - 1 t) + 3 v (w - 1 t)}{w^{2} - t^{2}} \cdot \frac{6 w + 6 t + u w + t u}{36 - u^{2}}$

Step 1.2

Factor the polynomial by factoring out the greatest common factor, $w - 1 t$ .

$\frac{(w - 1 t) (u + 3 v)}{w^{2} - t^{2}} \cdot \frac{6 w + 6 t + u w + t u}{36 - u^{2}}$

Step 1.3

Rewrite $- 1 t$ as $- t$ .

$\frac{(w - t) (u + 3 v)}{w^{2} - t^{2}} \cdot \frac{6 w + 6 t + u w + t u}{36 - u^{2}}$

Step 2

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

$\frac{(w - t) (u + 3 v)}{(w + t) (w - t)} \cdot \frac{6 w + 6 t + u w + t u}{36 - u^{2}}$

Step 3

Simplify the numerator.

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Step 3.1

Factor out the greatest common factor from each group.

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Step 3.1.1

Group the first two terms and the last two terms.

$\frac{(w - t) (u + 3 v)}{(w + t) (w - t)} \cdot \frac{(6 w + 6 t) + u w + t u}{36 - u^{2}}$

Step 3.1.2

Factor out the greatest common factor (GCF) from each group.

$\frac{(w - t) (u + 3 v)}{(w + t) (w - t)} \cdot \frac{6 (w + t) + u (w + t)}{36 - u^{2}}$

Step 3.2

Factor the polynomial by factoring out the greatest common factor, $w + t$ .

$\frac{(w - t) (u + 3 v)}{(w + t) (w - t)} \cdot \frac{(w + t) (6 + u)}{36 - u^{2}}$

Step 4

Simplify the denominator.

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Step 4.1

Rewrite $36$ as $6^{2}$ .

$\frac{(w - t) (u + 3 v)}{(w + t) (w - t)} \cdot \frac{(w + t) (6 + u)}{6^{2} - u^{2}}$

Step 4.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 6$ and $b = u$ .

$\frac{(w - t) (u + 3 v)}{(w + t) (w - t)} \cdot \frac{(w + t) (6 + u)}{(6 + u) (6 - u)}$

Step 5

Simplify terms.

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Step 5.1

Cancel the common factor of $w + t$ .

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Step 5.1.1

Cancel the common factor.

$\frac{(w - t) (u + 3 v)}{(w + t) (w - t)} \cdot \frac{(w + t) (6 + u)}{(6 + u) (6 - u)}$

Step 5.1.2

Rewrite the expression.

$\frac{(w - t) (u + 3 v)}{w - t} \cdot \frac{6 + u}{(6 + u) (6 - u)}$

Step 5.2

Multiply $\frac{(w - t) (u + 3 v)}{w - t}$ by $\frac{6 + u}{(6 + u) (6 - u)}$ .

$\frac{(w - t) (u + 3 v) (6 + u)}{(w - t) ((6 + u) (6 - u))}$

Step 5.3

Cancel the common factor of $w - t$ .

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Step 5.3.1

Cancel the common factor.

$\frac{(w - t) (u + 3 v) (6 + u)}{(w - t) ((6 + u) (6 - u))}$

Step 5.3.2

Rewrite the expression.

$\frac{(u + 3 v) (6 + u)}{(6 + u) (6 - u)}$

Step 5.4

Cancel the common factor of $6 + u$ .

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Step 5.4.1

Cancel the common factor.

$\frac{(u + 3 v) (6 + u)}{(6 + u) (6 - u)}$

Step 5.4.2

Rewrite the expression.

$\frac{u + 3 v}{6 - u}$