Algebra Examples

${(a + b)}^{3} - 8 b^{3}$

Step 1

Rewrite $8 b^{3}$ as ${(2 b)}^{3}$ .

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

Step 2

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = a + b$ and $b = 2 b$ .

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

Step 3

Simplify.

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Step 3.1

Multiply $2$ by $- 1$ .

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

Step 3.2

Subtract $2 b$ from $b$ .

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

Step 3.3

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

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

Step 3.4

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

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Step 3.4.1

Apply the distributive property.

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

Step 3.4.2

Apply the distributive property.

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

Step 3.4.3

Apply the distributive property.

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

Step 3.5

Simplify and combine like terms.

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Step 3.5.1

Simplify each term.

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Step 3.5.1.1

Multiply $a$ by $a$ .

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

Step 3.5.1.2

Multiply $b$ by $b$ .

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

Step 3.5.2

Add $a b$ and $b a$ .

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Step 3.5.2.1

Reorder $b$ and $a$ .

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

Step 3.5.2.2

Add $a b$ and $a b$ .

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

Step 3.6

Apply the distributive property.

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

Step 3.7

Rewrite using the commutative property of multiplication.

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

Step 3.8

Rewrite using the commutative property of multiplication.

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

Step 3.9

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

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Step 3.9.1

Move $b$ .

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

Step 3.9.2

Multiply $b$ by $b$ .

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

Step 3.10

Apply the product rule to $2 b$ .

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

Step 3.11

Raise $2$ to the power of $2$ .

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

Step 3.12

Add $2 a b$ and $2 a b$ .

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

Step 3.13

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

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

Step 3.14

Add $3 b^{2}$ and $4 b^{2}$ .

$(a - b) (a^{2} + 7 b^{2} + 4 a b)$