Algebra Examples

$\frac{a^{2} - b^{2}}{2 a + 2 b} \cdot \frac{a^{2} - a b}{{(b - a)}^{2}}$

Step 1

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

$\frac{(a + b) (a - b)}{2 a + 2 b} \cdot \frac{a^{2} - a b}{{(b - a)}^{2}}$

Step 2

Factor $2$ out of $2 a + 2 b$ .

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Step 2.1

Factor $2$ out of $2 a$ .

$\frac{(a + b) (a - b)}{2 (a) + 2 b} \cdot \frac{a^{2} - a b}{{(b - a)}^{2}}$

Step 2.2

Factor $2$ out of $2 b$ .

$\frac{(a + b) (a - b)}{2 (a) + 2 (b)} \cdot \frac{a^{2} - a b}{{(b - a)}^{2}}$

Step 2.3

Factor $2$ out of $2 (a) + 2 (b)$ .

$\frac{(a + b) (a - b)}{2 (a + b)} \cdot \frac{a^{2} - a b}{{(b - a)}^{2}}$

Step 3

Simplify the numerator.

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Step 3.1

Factor $a$ out of $a^{2} - a b$ .

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Step 3.1.1

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

$\frac{(a + b) (a - b)}{2 (a + b)} \cdot \frac{a \cdot a - a b}{{(b - a)}^{2}}$

Step 3.1.2

Factor $a$ out of $- a b$ .

$\frac{(a + b) (a - b)}{2 (a + b)} \cdot \frac{a \cdot a + a (- 1 b)}{{(b - a)}^{2}}$

Step 3.1.3

Factor $a$ out of $a \cdot a + a (- 1 b)$ .

$\frac{(a + b) (a - b)}{2 (a + b)} \cdot \frac{a (a - 1 b)}{{(b - a)}^{2}}$

Step 3.2

Rewrite $- 1 b$ as $- b$ .

$\frac{(a + b) (a - b)}{2 (a + b)} \cdot \frac{a (a - b)}{{(b - a)}^{2}}$

Step 4

Simplify terms.

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Step 4.1

Combine.

$\frac{(a + b) (a - b) (a (a - b))}{2 (a + b) {(b - a)}^{2}}$

Step 4.2

Cancel the common factor of $a + b$ .

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Step 4.2.1

Cancel the common factor.

$\frac{(a + b) (a - b) (a (a - b))}{2 (a + b) {(b - a)}^{2}}$

Step 4.2.2

Rewrite the expression.

$\frac{(a - b) (a (a - b))}{2 {(b - a)}^{2}}$

Step 4.3

Cancel the common factor of $a - b$ and ${(b - a)}^{2}$ .

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Step 4.3.1

Factor $- 1$ out of $a$ .

$\frac{(- 1 (- a) - b) (a (a - b))}{2 {(b - a)}^{2}}$

Step 4.3.2

Factor $- 1$ out of $- b$ .

$\frac{(- 1 (- a) - (b)) (a (a - b))}{2 {(b - a)}^{2}}$

Step 4.3.3

Factor $- 1$ out of $- 1 (- a) - (b)$ .

$\frac{- 1 (- a + b) (a (a - b))}{2 {(b - a)}^{2}}$

Step 4.3.4

Reorder terms.

$\frac{- 1 (- a + b) (a (a - b))}{2 {(- a + b)}^{2}}$

Step 4.3.5

Factor $- a + b$ out of $- 1 (- a + b) (a (a - b))$ .

$\frac{(- a + b) (- 1 (a (a - b)))}{2 {(- a + b)}^{2}}$

Step 4.3.6

Cancel the common factors.

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Step 4.3.6.1

Factor $- a + b$ out of $2 {(- a + b)}^{2}$ .

$\frac{(- a + b) (- 1 (a (a - b)))}{(- a + b) (2 (- a + b))}$

Step 4.3.6.2

Cancel the common factor.

$\frac{(- a + b) (- 1 (a (a - b)))}{(- a + b) (2 (- a + b))}$

Step 4.3.6.3

Rewrite the expression.

$\frac{- 1 (a (a - b))}{2 (- a + b)}$

Step 4.4

Cancel the common factor of $a - b$ and $- a + b$ .

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Step 4.4.1

Factor $- 1$ out of $a$ .

$\frac{- 1 (a (- 1 (- a) - b))}{2 (- a + b)}$

Step 4.4.2

Factor $- 1$ out of $- b$ .

$\frac{- 1 (a (- 1 (- a) - (b)))}{2 (- a + b)}$

Step 4.4.3

Factor $- 1$ out of $- 1 (- a) - (b)$ .

$\frac{- 1 (a (- 1 (- a + b)))}{2 (- a + b)}$

Step 4.4.4

Cancel the common factor.

$\frac{- 1 a (- 1 (- a + b))}{2 (- a + b)}$

Step 4.4.5

Rewrite the expression.

$\frac{- 1 a \cdot (- 1)}{2}$

Step 5

Simplify the numerator.

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Step 5.1

Multiply $- 1$ by $- 1$ .

$\frac{1 a}{2}$

Step 5.2

Multiply $a$ by $1$ .

$\frac{a}{2}$