Algebra Examples

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

Step 1

Rewrite the division as a fraction.

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

Step 2

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

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Step 2.1

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

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

Step 2.2

Combine.

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

Step 3

Apply the distributive property.

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

Step 4

Simplify by cancelling.

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Step 4.1

Cancel the common factor of $b$ .

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Step 4.1.1

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

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

Step 4.1.2

Cancel the common factor.

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

Step 4.1.3

Rewrite the expression.

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

Add $1$ and $1$ .

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

Step 4.6

Cancel the common factor of $a + b$ .

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Step 4.6.1

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

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

Step 4.6.2

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

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

Step 4.6.3

Cancel the common factor.

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

Step 4.6.4

Rewrite the expression.

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

Step 4.7

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

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

Step 4.8

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

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

Step 4.9

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

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

Step 4.10

Add $1$ and $1$ .

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

Step 4.11

Cancel the common factor of $a$ .

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Step 4.11.1

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

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

Step 4.11.2

Cancel the common factor.

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

Step 4.11.3

Rewrite the expression.

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

Step 4.12

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

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

Step 4.13

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

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

Step 4.14

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

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

Step 4.15

Add $1$ and $1$ .

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

Step 4.16

Cancel the common factor of $a + b$ .

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Step 4.16.1

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

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

Step 4.16.2

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

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

Step 4.16.3

Cancel the common factor.

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

Step 4.16.4

Rewrite the expression.

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

Step 4.17

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

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

Step 4.18

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

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

Step 4.19

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

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

Step 4.20

Add $1$ and $1$ .

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

Step 5

Simplify the numerator.

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Step 5.1

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

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Step 5.1.1

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

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

Step 5.1.2

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

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

Step 5.1.3

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

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

Step 5.2

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

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

Step 5.3

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

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Step 5.3.1

Apply the distributive property.

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

Step 5.3.2

Apply the distributive property.

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

Step 5.3.3

Apply the distributive property.

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

Step 5.4

Simplify and combine like terms.

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Step 5.4.1

Simplify each term.

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Step 5.4.1.1

Multiply $a$ by $a$ .

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

Step 5.4.1.2

Multiply $b$ by $b$ .

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

Step 5.4.2

Add $a b$ and $b a$ .

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Step 5.4.2.1

Reorder $b$ and $a$ .

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

Step 5.4.2.2

Add $a b$ and $a b$ .

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

Step 5.5

Rewrite using the commutative property of multiplication.

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

Step 5.6

Subtract $b a$ from $2 a b$ .

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Step 5.6.1

Move $b$ .

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

Step 5.6.2

Subtract $a b$ from $2 a b$ .

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

Step 5.7

Multiply $a$ by $1$ .

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

Step 6

Simplify the denominator.

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Step 6.1

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

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Step 6.1.1

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

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

Step 6.1.2

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

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

Step 6.1.3

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

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

Step 6.2

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

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

Step 6.3

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

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Step 6.3.1

Apply the distributive property.

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

Step 6.3.2

Apply the distributive property.

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

Step 6.3.3

Apply the distributive property.

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

Step 6.4

Simplify and combine like terms.

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Step 6.4.1

Simplify each term.

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Step 6.4.1.1

Multiply $a$ by $a$ .

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

Step 6.4.1.2

Multiply $b$ by $b$ .

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

Step 6.4.2

Add $a b$ and $b a$ .

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Step 6.4.2.1

Reorder $b$ and $a$ .

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

Step 6.4.2.2

Add $a b$ and $a b$ .

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

Step 6.5

Rewrite using the commutative property of multiplication.

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

Step 6.6

Subtract $a b$ from $2 a b$ .

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

Step 6.7

Multiply $a$ by $1$ .

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

Step 7

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

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Step 7.1

Cancel the common factor.

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

Step 7.2

Rewrite the expression.

$\frac{a}{b}$