Algebra Examples

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

Step 1

Rewrite the division as a fraction.

$\frac{a + b - \frac{2 a b}{a + b}}{\frac{a - b}{a + b} + \frac{b}{a}}$

Step 2

Multiply the numerator and denominator of the fraction by $(a + b) a$ .

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Step 2.1

Multiply $\frac{a + b - \frac{2 a b}{a + b}}{\frac{a - b}{a + b} + \frac{b}{a}}$ by $\frac{(a + b) a}{(a + b) a}$ .

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

Step 2.2

Combine.

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

Step 3

Apply the distributive property.

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

Step 4

Simplify by cancelling.

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Step 4.1

Cancel the common factor of $a + b$ .

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Step 4.1.1

Move the leading negative in $- \frac{2 a b}{a + b}$ into the numerator.

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

Step 4.1.2

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

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

Step 4.1.3

Cancel the common factor.

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

Step 4.1.4

Rewrite the expression.

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

Step 4.2

Raise $a$ to the power of $1$ .

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

Step 4.3

Raise $a$ to the power of $1$ .

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

Step 4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 4.5

Add $1$ and $1$ .

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

Step 4.6

Cancel the common factor of $a + b$ .

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Step 4.6.1

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

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

Step 4.6.2

Cancel the common factor.

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

Step 4.6.3

Rewrite the expression.

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

Step 4.7

Cancel the common factor of $a$ .

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Step 4.7.1

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

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

Step 4.7.2

Cancel the common factor.

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

Step 4.7.3

Rewrite the expression.

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

Step 5

Simplify the numerator.

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Step 5.1

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

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Step 5.1.1

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

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

Step 5.1.2

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

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

Step 5.1.3

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

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

Step 5.1.4

Factor $a^{2}$ out of $a^{2} ((a + b) a^{- 1} a) + a^{2} ((a + b) a^{- 1} b)$ .

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

Step 5.1.5

Factor $a^{2}$ out of $a^{2} ((a + b) a^{- 1} a + (a + b) a^{- 1} b) + a^{2} (- 2 b)$ .

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

Step 5.2

Factor by grouping.

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Step 5.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 2 \cdot 1 = - 2$ and whose sum is $b = 1$ .

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Step 5.2.1.1

Reorder terms.

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

Step 5.2.1.2

Reorder $a \cdot a^{- 1} (a + b)$ and $b a^{- 1} (a + b)$ .

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

Step 5.2.1.3

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

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

Step 5.2.1.4

Rewrite $1$ as $- 1$ plus $2$

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

Step 5.2.1.5

Apply the distributive property.

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

Step 5.2.1.6

Move parentheses.

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

Step 5.2.2

Combine exponents.

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Step 5.2.2.1

Multiply $a$ by $a^{- 1}$ by adding the exponents.

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Step 5.2.2.1.1

Multiply $a$ by $a^{- 1}$ .

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Step 5.2.2.1.1.1

Raise $a$ to the power of $1$ .

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

Step 5.2.2.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 5.2.2.1.2

Subtract $1$ from $1$ .

$\frac{a^{2} (- 2 b - 1 b a^{- 1} (a + b) + 2 b a^{- 1} (a + b) + a^{0} (a + b))}{a (a - b) + (a + b) b}$

Step 5.2.2.2

Simplify $a^{0}$ .

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

Step 5.2.2.3

Multiply $a + b$ by $1$ .

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

Step 5.2.3

Add $- 2 b$ and $b$ .

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

Step 5.2.4

Add $- 1 b a^{- 1} (a + b)$ and $2 b a^{- 1} (a + b)$ .

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

Step 5.2.5

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

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Step 5.2.5.1

Reorder the expression.

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Step 5.2.5.1.1

Move $b$ .

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

Step 5.2.5.1.2

Move $- b$ .

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

Step 5.2.5.2

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

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

Step 5.2.5.3

Factor $- 1$ out of $a$ .

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

Step 5.2.5.4

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

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

Step 5.2.5.5

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

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

Step 5.2.5.6

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

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

Step 5.2.6

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 5.2.7

Combine $b$ and $\frac{1}{a}$ .

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

Step 5.2.8

Apply the distributive property.

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

Step 5.2.9

Cancel the common factor of $a$ .

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Step 5.2.9.1

Move the leading negative in $- \frac{b}{a}$ into the numerator.

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

Step 5.2.9.2

Cancel the common factor.

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

Step 5.2.9.3

Rewrite the expression.

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

Step 5.2.10

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

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Step 5.2.10.1

Combine $b$ and $\frac{b}{a}$ .

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

Step 5.2.10.2

Raise $b$ to the power of $1$ .

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

Step 5.2.10.3

Raise $b$ to the power of $1$ .

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

Step 5.2.10.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 5.2.10.5

Add $1$ and $1$ .

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

Step 5.2.11

Add $- b$ and $b$ .

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

Step 5.2.12

Add $- \frac{b^{2}}{a} - a$ and $0$ .

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

Step 5.3

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

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

Step 5.4

Combine $- a$ and $\frac{a}{a}$ .

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

Step 5.5

Combine the numerators over the common denominator.

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

Step 5.6

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

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Step 5.6.1

Move $a$ .

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

Step 5.6.2

Multiply $a$ by $a$ .

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

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

Step 5.7

Combine exponents.

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Step 5.7.1

Factor out negative.

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

Step 5.7.2

Combine $a^{2}$ and $\frac{- b^{2} - a^{2}}{a}$ .

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

Step 5.8

Cancel the common factor of $a^{2}$ and $a$ .

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Step 5.8.1

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

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

Step 5.8.2

Cancel the common factors.

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Step 5.8.2.1

Raise $a$ to the power of $1$ .

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

Step 5.8.2.2

Factor $a$ out of $a^{1}$ .

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

Step 5.8.2.3

Cancel the common factor.

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

Step 5.8.2.4

Rewrite the expression.

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

Step 5.8.2.5

Divide $a (- b^{2} - a^{2})$ by $1$ .

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

Step 5.9

Apply the distributive property.

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

Step 5.10

Rewrite using the commutative property of multiplication.

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

Step 5.11

Rewrite using the commutative property of multiplication.

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

Step 5.12

Multiply $a$ by $a^{2}$ by adding the exponents.

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Step 5.12.1

Move $a^{2}$ .

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

Step 5.12.2

Multiply $a^{2}$ by $a$ .

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Step 5.12.2.1

Raise $a$ to the power of $1$ .

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

Step 5.12.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 5.12.3

Add $2$ and $1$ .

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

Step 5.13

Factor $a$ out of $- a b^{2} - a^{3}$ .

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Step 5.13.1

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

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

Step 5.13.2

Factor $a$ out of $- a^{3}$ .

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

Step 5.13.3

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

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

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

Step 6

Simplify the denominator.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Multiply $a$ by $a$ .

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

Step 6.3

Rewrite using the commutative property of multiplication.

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

Step 6.4

Apply the distributive property.

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

Step 6.5

Multiply $b$ by $b$ .

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

Step 6.6

Add $- a b$ and $a b$ .

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

Step 6.7

Add $a^{2}$ and $0$ .

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

Step 7

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

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Step 7.1

Factor $- 1$ out of $- b^{2}$ .

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

Step 7.2

Factor $- 1$ out of $- a^{2}$ .

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

Step 7.3

Factor $- 1$ out of $- (b^{2}) - (a^{2})$ .

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

Step 7.4

Rewrite $- (b^{2} + a^{2})$ as $- 1 (b^{2} + a^{2})$ .

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

Step 7.5

Reorder terms.

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

Step 7.6

Cancel the common factor.

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

Step 7.7

Divide $- a \cdot (- 1)$ by $1$ .

$- a \cdot (- 1)$

Step 8

Multiply $- a (- 1)$ .

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Step 8.1

Multiply $- 1$ by $- 1$ .

$1 a$

Step 8.2

Multiply $a$ by $1$ .

$a$