Algebra Examples

$(\frac{a}{a - b} - \frac{a - b}{a + b} + \frac{b^{2}}{a^{2} - b^{2}}) \cdot \frac{a - b}{3 a b}$

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 $b = b$ .

$(\frac{a}{a - b} - \frac{a - b}{a + b} + \frac{b^{2}}{(a + b) (a - b)}) \cdot \frac{a - b}{3 a b}$

Step 2

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

$(\frac{a}{a - b} \cdot \frac{a + b}{a + b} - \frac{a - b}{a + b} + \frac{b^{2}}{(a + b) (a - b)}) \cdot \frac{a - b}{3 a b}$

Step 3

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

$(\frac{a}{a - b} \cdot \frac{a + b}{a + b} - \frac{a - b}{a + b} \cdot \frac{a - b}{a - b} + \frac{b^{2}}{(a + b) (a - b)}) \cdot \frac{a - b}{3 a b}$

Step 4

Write each expression with a common denominator of $(a - b) (a + b)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{a}{a - b}$ by $\frac{a + b}{a + b}$ .

$(\frac{a (a + b)}{(a - b) (a + b)} - \frac{a - b}{a + b} \cdot \frac{a - b}{a - b} + \frac{b^{2}}{(a + b) (a - b)}) \cdot \frac{a - b}{3 a b}$

Step 4.2

Multiply $\frac{a - b}{a + b}$ by $\frac{a - b}{a - b}$ .

$(\frac{a (a + b)}{(a - b) (a + b)} - \frac{(a - b) (a - b)}{(a + b) (a - b)} + \frac{b^{2}}{(a + b) (a - b)}) \cdot \frac{a - b}{3 a b}$

Step 4.3

Reorder the factors of $(a - b) (a + b)$ .

$(\frac{a (a + b)}{(a + b) (a - b)} - \frac{(a - b) (a - b)}{(a + b) (a - b)} + \frac{b^{2}}{(a + b) (a - b)}) \cdot \frac{a - b}{3 a b}$

Step 5

Simplify terms.

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Step 5.1

Combine the numerators over the common denominator.

$(\frac{a (a + b) - (a - b) (a - b)}{(a + b) (a - b)} + \frac{b^{2}}{(a + b) (a - b)}) \cdot \frac{a - b}{3 a b}$

Step 5.2

Combine the numerators over the common denominator.

$\frac{a (a + b) - (a - b) (a - b) + b^{2}}{(a + b) (a - b)} \cdot \frac{a - b}{3 a b}$

Step 6

Simplify each term.

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Step 6.1

Apply the distributive property.

$\frac{a \cdot a + a b - (a - b) (a - b) + b^{2}}{(a + b) (a - b)} \cdot \frac{a - b}{3 a b}$

Step 6.2

Multiply $a$ by $a$ .

$\frac{a^{2} + a b - (a - b) (a - b) + b^{2}}{(a + b) (a - b)} \cdot \frac{a - b}{3 a b}$

Step 6.3

Apply the distributive property.

$\frac{a^{2} + a b + (- a - - b) (a - b) + b^{2}}{(a + b) (a - b)} \cdot \frac{a - b}{3 a b}$

Step 6.4

Multiply $- - b$ .

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Step 6.4.1

Multiply $- 1$ by $- 1$ .

$\frac{a^{2} + a b + (- a + 1 b) (a - b) + b^{2}}{(a + b) (a - b)} \cdot \frac{a - b}{3 a b}$

Step 6.4.2

Multiply $b$ by $1$ .

$\frac{a^{2} + a b + (- a + b) (a - b) + b^{2}}{(a + b) (a - b)} \cdot \frac{a - b}{3 a b}$

Step 6.5

Expand $(- a + b) (a - b)$ using the FOIL Method.

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Step 6.5.1

Apply the distributive property.

$\frac{a^{2} + a b - a (a - b) + b (a - b) + b^{2}}{(a + b) (a - b)} \cdot \frac{a - b}{3 a b}$

Step 6.5.2

Apply the distributive property.

$\frac{a^{2} + a b - a \cdot a - a (- b) + b (a - b) + b^{2}}{(a + b) (a - b)} \cdot \frac{a - b}{3 a b}$

Step 6.5.3

Apply the distributive property.

$\frac{a^{2} + a b - a \cdot a - a (- b) + b a + b (- b) + b^{2}}{(a + b) (a - b)} \cdot \frac{a - b}{3 a b}$

Step 6.6

Simplify and combine like terms.

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Step 6.6.1

Simplify each term.

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Step 6.6.1.1

Multiply $a$ by $a$ by adding the exponents.

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Step 6.6.1.1.1

Move $a$ .

$\frac{a^{2} + a b - (a \cdot a) - a (- b) + b a + b (- b) + b^{2}}{(a + b) (a - b)} \cdot \frac{a - b}{3 a b}$

Step 6.6.1.1.2

Multiply $a$ by $a$ .

$\frac{a^{2} + a b - a^{2} - a (- b) + b a + b (- b) + b^{2}}{(a + b) (a - b)} \cdot \frac{a - b}{3 a b}$

Step 6.6.1.2

Rewrite using the commutative property of multiplication.

$\frac{a^{2} + a b - a^{2} - 1 \cdot - 1 a b + b a + b (- b) + b^{2}}{(a + b) (a - b)} \cdot \frac{a - b}{3 a b}$

Step 6.6.1.3

Multiply $- 1$ by $- 1$ .

$\frac{a^{2} + a b - a^{2} + 1 a b + b a + b (- b) + b^{2}}{(a + b) (a - b)} \cdot \frac{a - b}{3 a b}$

Step 6.6.1.4

Multiply $a$ by $1$ .

$\frac{a^{2} + a b - a^{2} + a b + b a + b (- b) + b^{2}}{(a + b) (a - b)} \cdot \frac{a - b}{3 a b}$

Step 6.6.1.5

Rewrite using the commutative property of multiplication.

$\frac{a^{2} + a b - a^{2} + a b + b a - b \cdot b + b^{2}}{(a + b) (a - b)} \cdot \frac{a - b}{3 a b}$

Step 6.6.1.6

Multiply $b$ by $b$ by adding the exponents.

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Step 6.6.1.6.1

Move $b$ .

$\frac{a^{2} + a b - a^{2} + a b + b a - (b \cdot b) + b^{2}}{(a + b) (a - b)} \cdot \frac{a - b}{3 a b}$

Step 6.6.1.6.2

Multiply $b$ by $b$ .

$\frac{a^{2} + a b - a^{2} + a b + b a - b^{2} + b^{2}}{(a + b) (a - b)} \cdot \frac{a - b}{3 a b}$

Step 6.6.2

Add $a b$ and $b a$ .

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Step 6.6.2.1

Reorder $b$ and $a$ .

$\frac{a^{2} + a b - a^{2} + a b + a b - b^{2} + b^{2}}{(a + b) (a - b)} \cdot \frac{a - b}{3 a b}$

Step 6.6.2.2

Add $a b$ and $a b$ .

$\frac{a^{2} + a b - a^{2} + 2 a b - b^{2} + b^{2}}{(a + b) (a - b)} \cdot \frac{a - b}{3 a b}$

Step 7

Simplify terms.

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Step 7.1

Combine the opposite terms in $a^{2} + a b - a^{2} + 2 a b - b^{2} + b^{2}$ .

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Step 7.1.1

Subtract $a^{2}$ from $a^{2}$ .

$\frac{a b + 0 + 2 a b - b^{2} + b^{2}}{(a + b) (a - b)} \cdot \frac{a - b}{3 a b}$

Step 7.1.2

Add $a b$ and $0$ .

$\frac{a b + 2 a b - b^{2} + b^{2}}{(a + b) (a - b)} \cdot \frac{a - b}{3 a b}$

Step 7.1.3

Add $- b^{2}$ and $b^{2}$ .

$\frac{a b + 2 a b + 0}{(a + b) (a - b)} \cdot \frac{a - b}{3 a b}$

Step 7.1.4

Add $a b + 2 a b$ and $0$ .

$\frac{a b + 2 a b}{(a + b) (a - b)} \cdot \frac{a - b}{3 a b}$

Step 7.2

Add $a b$ and $2 a b$ .

$\frac{3 a b}{(a + b) (a - b)} \cdot \frac{a - b}{3 a b}$

Step 7.3

Cancel the common factor of $3 a b$ .

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Step 7.3.1

Cancel the common factor.

$\frac{3 a b}{(a + b) (a - b)} \cdot \frac{a - b}{3 a b}$

Step 7.3.2

Rewrite the expression.

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

Step 7.4

Cancel the common factor of $a - b$ .

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Step 7.4.1

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

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

Step 7.4.2

Cancel the common factor.

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

Step 7.4.3

Rewrite the expression.

$\frac{1}{a + b}$