Algebra Examples

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

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

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

Tap for more steps...

Step 1.1.1

Factor $b$ out of $a b$ .

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

Step 1.1.2

Factor $b$ out of $- b$ .

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

Step 1.1.3

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

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

Step 1.2

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

Tap for more steps...

Step 1.2.1

Factor $a$ out of $a b$ .

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

Step 1.2.2

Factor $a$ out of $- a$ .

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

Step 1.2.3

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

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

Step 1.3

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 $b = b$ .

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

Step 2

To write $\frac{b (a - 1)}{a}$ as a fraction with a common denominator, multiply by $\frac{b}{b}$ .

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

Step 3

To write $- \frac{a (b - 1)}{b}$ as a fraction with a common denominator, multiply by $\frac{a}{a}$ .

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

Step 4

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

Tap for more steps...

Step 4.1

Multiply $\frac{b (a - 1)}{a}$ by $\frac{b}{b}$ .

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

Step 4.2

Multiply $\frac{a (b - 1)}{b}$ by $\frac{a}{a}$ .

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

Step 4.3

Reorder the factors of $b a$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

Tap for more steps...

Step 7.1

Simplify each term.

Tap for more steps...

Step 7.1.1

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

Tap for more steps...

Step 7.1.1.1

Move $b$ .

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

Step 7.1.1.2

Multiply $b$ by $b$ .

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

Step 7.1.2

Apply the distributive property.

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

Step 7.1.3

Move $- 1$ to the left of $b^{2}$ .

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

Step 7.1.4

Rewrite $- 1 b^{2}$ as $- b^{2}$ .

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

Step 7.1.5

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

Tap for more steps...

Step 7.1.5.1

Move $a$ .

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

Step 7.1.5.2

Multiply $a$ by $a$ .

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

Step 7.1.6

Apply the distributive property.

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

Step 7.1.7

Multiply $- a^{2} \cdot - 1$ .

Tap for more steps...

Step 7.1.7.1

Multiply $- 1$ by $- 1$ .

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

Step 7.1.7.2

Multiply $a^{2}$ by $1$ .

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

Step 7.1.8

Apply the distributive property.

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

Step 7.1.9

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

Tap for more steps...

Step 7.1.9.1

Apply the distributive property.

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

Step 7.1.9.2

Apply the distributive property.

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

Step 7.1.9.3

Apply the distributive property.

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

Step 7.1.10

Simplify and combine like terms.

Tap for more steps...

Step 7.1.10.1

Simplify each term.

Tap for more steps...

Step 7.1.10.1.1

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

Tap for more steps...

Step 7.1.10.1.1.1

Move $a$ .

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

Step 7.1.10.1.1.2

Multiply $a$ by $a$ .

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

Step 7.1.10.1.2

Rewrite using the commutative property of multiplication.

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

Step 7.1.10.1.3

Multiply $- 1$ by $- 1$ .

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

Step 7.1.10.1.4

Multiply $a$ by $1$ .

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

Step 7.1.10.1.5

Rewrite using the commutative property of multiplication.

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

Step 7.1.10.1.6

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

Tap for more steps...

Step 7.1.10.1.6.1

Move $b$ .

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

Step 7.1.10.1.6.2

Multiply $b$ by $b$ .

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

Step 7.1.10.1.7

Multiply $- 1$ by $- 1$ .

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

Step 7.1.10.1.8

Multiply $b^{2}$ by $1$ .

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

Step 7.1.10.2

Subtract $b a$ from $a b$ .

Tap for more steps...

Step 7.1.10.2.1

Move $b$ .

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

Step 7.1.10.2.2

Subtract $a b$ from $a b$ .

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

Step 7.1.10.3

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

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

Step 7.2

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

Tap for more steps...

Step 7.2.1

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

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

Step 7.2.2

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

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

Step 7.2.3

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

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

Step 7.2.4

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

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

Step 8

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

Tap for more steps...

Step 8.1

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

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

Step 8.2

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

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

Step 8.3

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

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

Step 9

Cancel the common factor of $b$ .

Tap for more steps...

Step 9.1

Cancel the common factor.

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

Step 9.2

Rewrite the expression.

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

Step 10

Cancel the common factor of $a$ .

Tap for more steps...

Step 10.1

Cancel the common factor.

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

Step 10.2

Divide $b - a$ by $1$ .

$b - a$