Basic Math Examples

$\frac{8 v w}{v^{2} - 4 w^{2}} + \frac{v - 2 w}{v + 2 w}$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $4 w^{2}$ as ${(2 w)}^{2}$ .

$\frac{8 v w}{v^{2} - {(2 w)}^{2}} + \frac{v - 2 w}{v + 2 w}$

Step 1.2

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 $b = 2 w$ .

$\frac{8 v w}{(v + 2 w) (v - (2 w))} + \frac{v - 2 w}{v + 2 w}$

Step 1.3

Multiply $2$ by $- 1$ .

$\frac{8 v w}{(v + 2 w) (v - 2 w)} + \frac{v - 2 w}{v + 2 w}$

Step 2

To write $\frac{v - 2 w}{v + 2 w}$ as a fraction with a common denominator, multiply by $\frac{v - 2 w}{v - 2 w}$ .

$\frac{8 v w}{(v + 2 w) (v - 2 w)} + \frac{v - 2 w}{v + 2 w} \cdot \frac{v - 2 w}{v - 2 w}$

Step 3

Simplify terms.

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Step 3.1

Multiply $\frac{v - 2 w}{v + 2 w}$ by $\frac{v - 2 w}{v - 2 w}$ .

$\frac{8 v w}{(v + 2 w) (v - 2 w)} + \frac{(v - 2 w) (v - 2 w)}{(v + 2 w) (v - 2 w)}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{8 v w + (v - 2 w) (v - 2 w)}{(v + 2 w) (v - 2 w)}$

Step 4

Simplify the numerator.

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Step 4.1

Expand $(v - 2 w) (v - 2 w)$ using the FOIL Method.

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Step 4.1.1

Apply the distributive property.

$\frac{8 v w + v (v - 2 w) - 2 w (v - 2 w)}{(v + 2 w) (v - 2 w)}$

Step 4.1.2

Apply the distributive property.

$\frac{8 v w + v \cdot v + v (- 2 w) - 2 w (v - 2 w)}{(v + 2 w) (v - 2 w)}$

Step 4.1.3

Apply the distributive property.

$\frac{8 v w + v \cdot v + v (- 2 w) - 2 w v - 2 w (- 2 w)}{(v + 2 w) (v - 2 w)}$

Step 4.2

Simplify and combine like terms.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Multiply $v$ by $v$ .

$\frac{8 v w + v^{2} + v (- 2 w) - 2 w v - 2 w (- 2 w)}{(v + 2 w) (v - 2 w)}$

Step 4.2.1.2

Rewrite using the commutative property of multiplication.

$\frac{8 v w + v^{2} - 2 v w - 2 w v - 2 w (- 2 w)}{(v + 2 w) (v - 2 w)}$

Step 4.2.1.3

Rewrite using the commutative property of multiplication.

$\frac{8 v w + v^{2} - 2 v w - 2 w v - 2 \cdot - 2 w \cdot w}{(v + 2 w) (v - 2 w)}$

Step 4.2.1.4

Multiply $w$ by $w$ by adding the exponents.

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Step 4.2.1.4.1

Move $w$ .

$\frac{8 v w + v^{2} - 2 v w - 2 w v - 2 \cdot - 2 (w \cdot w)}{(v + 2 w) (v - 2 w)}$

Step 4.2.1.4.2

Multiply $w$ by $w$ .

$\frac{8 v w + v^{2} - 2 v w - 2 w v - 2 \cdot - 2 w^{2}}{(v + 2 w) (v - 2 w)}$

Step 4.2.1.5

Multiply $- 2$ by $- 2$ .

$\frac{8 v w + v^{2} - 2 v w - 2 w v + 4 w^{2}}{(v + 2 w) (v - 2 w)}$

Step 4.2.2

Subtract $2 w v$ from $- 2 v w$ .

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Step 4.2.2.1

Move $w$ .

$\frac{8 v w + v^{2} - 2 v w - 2 v w + 4 w^{2}}{(v + 2 w) (v - 2 w)}$

Step 4.2.2.2

Subtract $2 v w$ from $- 2 v w$ .

$\frac{8 v w + v^{2} - 4 v w + 4 w^{2}}{(v + 2 w) (v - 2 w)}$

Step 4.3

Subtract $4 v w$ from $8 v w$ .

$\frac{v^{2} + 4 v w + 4 w^{2}}{(v + 2 w) (v - 2 w)}$

Step 4.4

Factor using the perfect square rule.

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Step 4.4.1

Rewrite $4 w^{2}$ as ${(2 w)}^{2}$ .

$\frac{v^{2} + 4 v w + {(2 w)}^{2}}{(v + 2 w) (v - 2 w)}$

Step 4.4.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 v w = 2 \cdot v \cdot (2 w)$

Step 4.4.3

Rewrite the polynomial.

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

Step 4.4.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = v$ and $b = 2 w$ .

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

Step 5

Cancel the common factor of ${(v + 2 w)}^{2}$ and $v + 2 w$ .

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Step 5.1

Factor $v + 2 w$ out of ${(v + 2 w)}^{2}$ .

$\frac{(v + 2 w) (v + 2 w)}{(v + 2 w) (v - 2 w)}$

Step 5.2

Cancel the common factors.

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Step 5.2.1

Cancel the common factor.

$\frac{(v + 2 w) (v + 2 w)}{(v + 2 w) (v - 2 w)}$

Step 5.2.2

Rewrite the expression.

$\frac{v + 2 w}{v - 2 w}$