Basic Math Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Simplify.

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Step 3.1

Add $2 a$ and $a$ .

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

Step 3.2

Apply the product rule to $2 a$ .

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

Step 3.3

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

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

Step 3.4

Multiply $2$ by $- 1$ .

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

Step 3.5

Apply the distributive property.

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

Step 3.6

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

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Step 3.6.1

Move $a$ .

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

Step 3.6.2

Multiply $a$ by $a$ .

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

Step 3.7

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

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

Step 3.8

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

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Step 3.8.1

Apply the distributive property.

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

Step 3.8.2

Apply the distributive property.

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

Step 3.8.3

Apply the distributive property.

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

Step 3.9

Simplify and combine like terms.

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Step 3.9.1

Simplify each term.

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Step 3.9.1.1

Multiply $a$ by $a$ .

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

Step 3.9.1.2

Multiply $b$ by $b$ .

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

Step 3.9.2

Add $a b$ and $b a$ .

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Step 3.9.2.1

Reorder $b$ and $a$ .

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

Step 3.9.2.2

Add $a b$ and $a b$ .

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

Step 3.10

Subtract $2 a^{2}$ from $4 a^{2}$ .

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

Step 3.11

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

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

Step 3.12

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

$(3 a + b) (3 a^{2} + 0 + b^{2})$

Step 3.13

Add $3 a^{2}$ and $0$ .

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