Algebra Examples

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

Step 1

Expand

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

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

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Multiply

$a$ by

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

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Step 2.1.1.1

Multiply

$a$ by

$a^{2}$ .

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Step 2.1.1.1.1

Raise

$a$ to the power of

$1$ .

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

Step 2.1.1.1.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.1.1.2

Add

$1$ and

$2$ .

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

Step 2.1.2

Rewrite using the commutative property of multiplication.

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

Step 2.1.3

Multiply

$a$ by

$a$ by adding the exponents.

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Step 2.1.3.1

Move

$a$ .

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

Step 2.1.3.2

Multiply

$a$ by

$a$ .

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

Step 2.1.4

Rewrite using the commutative property of multiplication.

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

Step 2.1.5

Multiply

$b$ by

$b$ by adding the exponents.

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Step 2.1.5.1

Move

$b$ .

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

Step 2.1.5.2

Multiply

$b$ by

$b$ .

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

Step 2.1.6

Multiply

$b$ by

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

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Step 2.1.6.1

Multiply

$b$ by

$b^{2}$ .

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Step 2.1.6.1.1

Raise

$b$ to the power of

$1$ .

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

Step 2.1.6.1.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.1.6.2

Add

$1$ and

$2$ .

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

Step 2.2

Combine the opposite terms in

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

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Step 2.2.1

Reorder the factors in the terms

$- a^{2} b$ and

$b a^{2}$ .

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

Step 2.2.2

Add

$- a^{2} b$ and

$a^{2} b$ .

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

Step 2.2.3

Add

$a^{3} + a b^{2}$ and

$0$ .

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

Step 2.2.4

Reorder the factors in the terms

$a b^{2}$ and

$- b^{2} a$ .

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

Step 2.2.5

Subtract

$b^{2} a$ from

$b^{2} a$ .

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

Step 2.2.6

Add

$a^{3}$ and

$0$ .

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