Algebra Examples

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

Step 1

Expand $(a + b) (a^{3} - a^{2} b + a b^{2} - b^{3})$ by multiplying each term in the first expression by each term in the second expression.

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

Step 2

Simplify terms.

Tap for more steps...

Step 2.1

Simplify each term.

Tap for more steps...

Step 2.1.1

Multiply $a$ by $a^{3}$ by adding the exponents.

Tap for more steps...

Step 2.1.1.1

Multiply $a$ by $a^{3}$ .

Tap for more steps...

Step 2.1.1.1.1

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

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

Step 2.1.1.1.2

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

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

Step 2.1.1.2

Add $1$ and $3$ .

$a^{4} + a (- a^{2} b) + a (a b^{2}) + a (- b^{3}) + b a^{3} + b (- a^{2} b) + b (a b^{2}) + b (- b^{3})$

Step 2.1.2

Rewrite using the commutative property of multiplication.

$a^{4} - a (a^{2} b) + a (a b^{2}) + a (- b^{3}) + b a^{3} + b (- a^{2} b) + b (a b^{2}) + b (- b^{3})$

Step 2.1.3

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

Tap for more steps...

Step 2.1.3.1

Move $a^{2}$ .

$a^{4} - (a^{2} a) b + a (a b^{2}) + a (- b^{3}) + b a^{3} + b (- a^{2} b) + b (a b^{2}) + b (- b^{3})$

Step 2.1.3.2

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

Tap for more steps...

Step 2.1.3.2.1

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

$a^{4} - (a^{2} a^{1}) b + a (a b^{2}) + a (- b^{3}) + b a^{3} + b (- a^{2} b) + b (a b^{2}) + b (- b^{3})$

Step 2.1.3.2.2

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

$a^{4} - a^{2 + 1} b + a (a b^{2}) + a (- b^{3}) + b a^{3} + b (- a^{2} b) + b (a b^{2}) + b (- b^{3})$

Step 2.1.3.3

Add $2$ and $1$ .

$a^{4} - a^{3} b + a (a b^{2}) + a (- b^{3}) + b a^{3} + b (- a^{2} b) + b (a b^{2}) + b (- b^{3})$

Step 2.1.4

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

Tap for more steps...

Step 2.1.4.1

Move $a$ .

$a^{4} - a^{3} b + a \cdot a b^{2} + a (- b^{3}) + b a^{3} + b (- a^{2} b) + b (a b^{2}) + b (- b^{3})$

Step 2.1.4.2

Multiply $a$ by $a$ .

$a^{4} - a^{3} b + a^{2} b^{2} + a (- b^{3}) + b a^{3} + b (- a^{2} b) + b (a b^{2}) + b (- b^{3})$

Step 2.1.5

Rewrite using the commutative property of multiplication.

$a^{4} - a^{3} b + a^{2} b^{2} - a b^{3} + b a^{3} + b (- a^{2} b) + b (a b^{2}) + b (- b^{3})$

Step 2.1.6

Rewrite using the commutative property of multiplication.

$a^{4} - a^{3} b + a^{2} b^{2} - a b^{3} + b a^{3} - b (a^{2} b) + b (a b^{2}) + b (- b^{3})$

Step 2.1.7

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

Tap for more steps...

Step 2.1.7.1

Move $b$ .

$a^{4} - a^{3} b + a^{2} b^{2} - a b^{3} + b a^{3} - (b \cdot b) a^{2} + b (a b^{2}) + b (- b^{3})$

Step 2.1.7.2

Multiply $b$ by $b$ .

$a^{4} - a^{3} b + a^{2} b^{2} - a b^{3} + b a^{3} - b^{2} a^{2} + b (a b^{2}) + b (- b^{3})$

Step 2.1.8

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

Tap for more steps...

Step 2.1.8.1

Move $b^{2}$ .

$a^{4} - a^{3} b + a^{2} b^{2} - a b^{3} + b a^{3} - b^{2} a^{2} + b^{2} b a + b (- b^{3})$

Step 2.1.8.2

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

Tap for more steps...

Step 2.1.8.2.1

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

$a^{4} - a^{3} b + a^{2} b^{2} - a b^{3} + b a^{3} - b^{2} a^{2} + b^{2} b^{1} a + b (- b^{3})$

Step 2.1.8.2.2

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

$a^{4} - a^{3} b + a^{2} b^{2} - a b^{3} + b a^{3} - b^{2} a^{2} + b^{2 + 1} a + b (- b^{3})$

Step 2.1.8.3

Add $2$ and $1$ .

$a^{4} - a^{3} b + a^{2} b^{2} - a b^{3} + b a^{3} - b^{2} a^{2} + b^{3} a + b (- b^{3})$

Step 2.1.9

Rewrite using the commutative property of multiplication.

$a^{4} - a^{3} b + a^{2} b^{2} - a b^{3} + b a^{3} - b^{2} a^{2} + b^{3} a - b \cdot b^{3}$

Step 2.1.10

Multiply $b$ by $b^{3}$ by adding the exponents.

Tap for more steps...

Step 2.1.10.1

Move $b^{3}$ .

$a^{4} - a^{3} b + a^{2} b^{2} - a b^{3} + b a^{3} - b^{2} a^{2} + b^{3} a - (b^{3} b)$

Step 2.1.10.2

Multiply $b^{3}$ by $b$ .

Tap for more steps...

Step 2.1.10.2.1

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

$a^{4} - a^{3} b + a^{2} b^{2} - a b^{3} + b a^{3} - b^{2} a^{2} + b^{3} a - (b^{3} b^{1})$

Step 2.1.10.2.2

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

$a^{4} - a^{3} b + a^{2} b^{2} - a b^{3} + b a^{3} - b^{2} a^{2} + b^{3} a - b^{3 + 1}$

Step 2.1.10.3

Add $3$ and $1$ .

$a^{4} - a^{3} b + a^{2} b^{2} - a b^{3} + b a^{3} - b^{2} a^{2} + b^{3} a - b^{4}$

Step 2.2

Combine the opposite terms in $a^{4} - a^{3} b + a^{2} b^{2} - a b^{3} + b a^{3} - b^{2} a^{2} + b^{3} a - b^{4}$ .

Tap for more steps...

Step 2.2.1

Reorder the factors in the terms $- a^{3} b$ and $b a^{3}$ .

$a^{4} - a^{3} b + a^{2} b^{2} - a b^{3} + a^{3} b - b^{2} a^{2} + b^{3} a - b^{4}$

Step 2.2.2

Add $- a^{3} b$ and $a^{3} b$ .

$a^{4} + a^{2} b^{2} - a b^{3} + 0 - b^{2} a^{2} + b^{3} a - b^{4}$

Step 2.2.3

Add $a^{4} + a^{2} b^{2} - a b^{3}$ and $0$ .

$a^{4} + a^{2} b^{2} - a b^{3} - b^{2} a^{2} + b^{3} a - b^{4}$

Step 2.2.4

Reorder the factors in the terms $a^{2} b^{2}$ and $- b^{2} a^{2}$ .

$a^{4} + a^{2} b^{2} - a b^{3} - a^{2} b^{2} + b^{3} a - b^{4}$

Step 2.2.5

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

$a^{4} - a b^{3} + 0 + b^{3} a - b^{4}$

Step 2.2.6

Add $a^{4} - a b^{3}$ and $0$ .

$a^{4} - a b^{3} + b^{3} a - b^{4}$

Step 2.2.7

Reorder the factors in the terms $- a b^{3}$ and $b^{3} a$ .

$a^{4} - b^{3} a + b^{3} a - b^{4}$

Step 2.2.8

Add $- b^{3} a$ and $b^{3} a$ .

$a^{4} + 0 - b^{4}$

Step 2.2.9

Add $a^{4}$ and $0$ .

$a^{4} - b^{4}$