Basic Math Examples

$512 + {(3 k + 4)}^{3}$

Step 1

Rewrite $512$ as $8^{3}$ .

$8^{3} + {(3 k + 4)}^{3}$

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 $a = 8$ and $b = 3 k + 4$ .

$(8 + 3 k + 4) (8^{2} - 8 (3 k + 4) + {(3 k + 4)}^{2})$

Step 3

Simplify.

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Step 3.1

Add $8$ and $4$ .

$(3 k + 12) (8^{2} - 8 (3 k + 4) + {(3 k + 4)}^{2})$

Step 3.2

Factor $3$ out of $3 k + 12$ .

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Step 3.2.1

Factor $3$ out of $3 k$ .

$(3 (k) + 12) (8^{2} - 8 (3 k + 4) + {(3 k + 4)}^{2})$

Step 3.2.2

Factor $3$ out of $12$ .

$(3 k + 3 \cdot 4) (8^{2} - 8 (3 k + 4) + {(3 k + 4)}^{2})$

Step 3.2.3

Factor $3$ out of $3 k + 3 \cdot 4$ .

$3 (k + 4) (8^{2} - 8 (3 k + 4) + {(3 k + 4)}^{2})$

Step 3.3

Raise $8$ to the power of $2$ .

$3 (k + 4) (64 - 8 (3 k + 4) + {(3 k + 4)}^{2})$

Step 3.4

Apply the distributive property.

$3 (k + 4) (64 - 8 (3 k) - 8 \cdot 4 + {(3 k + 4)}^{2})$

Step 3.5

Multiply $3$ by $- 8$ .

$3 (k + 4) (64 - 24 k - 8 \cdot 4 + {(3 k + 4)}^{2})$

Step 3.6

Multiply $- 8$ by $4$ .

$3 (k + 4) (64 - 24 k - 32 + {(3 k + 4)}^{2})$

Step 3.7

Rewrite ${(3 k + 4)}^{2}$ as $(3 k + 4) (3 k + 4)$ .

$3 (k + 4) (64 - 24 k - 32 + (3 k + 4) (3 k + 4))$

Step 3.8

Expand $(3 k + 4) (3 k + 4)$ using the FOIL Method.

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Step 3.8.1

Apply the distributive property.

$3 (k + 4) (64 - 24 k - 32 + 3 k (3 k + 4) + 4 (3 k + 4))$

Step 3.8.2

Apply the distributive property.

$3 (k + 4) (64 - 24 k - 32 + 3 k (3 k) + 3 k \cdot 4 + 4 (3 k + 4))$

Step 3.8.3

Apply the distributive property.

$3 (k + 4) (64 - 24 k - 32 + 3 k (3 k) + 3 k \cdot 4 + 4 (3 k) + 4 \cdot 4)$

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

Rewrite using the commutative property of multiplication.

$3 (k + 4) (64 - 24 k - 32 + 3 \cdot 3 k \cdot k + 3 k \cdot 4 + 4 (3 k) + 4 \cdot 4)$

Step 3.9.1.2

Multiply $k$ by $k$ by adding the exponents.

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Step 3.9.1.2.1

Move $k$ .

$3 (k + 4) (64 - 24 k - 32 + 3 \cdot 3 (k \cdot k) + 3 k \cdot 4 + 4 (3 k) + 4 \cdot 4)$

Step 3.9.1.2.2

Multiply $k$ by $k$ .

$3 (k + 4) (64 - 24 k - 32 + 3 \cdot 3 k^{2} + 3 k \cdot 4 + 4 (3 k) + 4 \cdot 4)$

Step 3.9.1.3

Multiply $3$ by $3$ .

$3 (k + 4) (64 - 24 k - 32 + 9 k^{2} + 3 k \cdot 4 + 4 (3 k) + 4 \cdot 4)$

Step 3.9.1.4

Multiply $4$ by $3$ .

$3 (k + 4) (64 - 24 k - 32 + 9 k^{2} + 12 k + 4 (3 k) + 4 \cdot 4)$

Step 3.9.1.5

Multiply $3$ by $4$ .

$3 (k + 4) (64 - 24 k - 32 + 9 k^{2} + 12 k + 12 k + 4 \cdot 4)$

Step 3.9.1.6

Multiply $4$ by $4$ .

$3 (k + 4) (64 - 24 k - 32 + 9 k^{2} + 12 k + 12 k + 16)$

Step 3.9.2

Add $12 k$ and $12 k$ .

$3 (k + 4) (64 - 24 k - 32 + 9 k^{2} + 24 k + 16)$

Step 3.10

Subtract $32$ from $64$ .

$3 (k + 4) (- 24 k + 32 + 9 k^{2} + 24 k + 16)$

Step 3.11

Add $- 24 k$ and $24 k$ .

$3 (k + 4) (32 + 9 k^{2} + 0 + 16)$

Step 3.12

Add $32$ and $0$ .

$3 (k + 4) (9 k^{2} + 32 + 16)$

Step 3.13

Add $32$ and $16$ .

$3 (k + 4) (9 k^{2} + 48)$

Step 3.14

Factor.

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Step 3.14.1

Factor $3$ out of $9 k^{2} + 48$ .

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Step 3.14.1.1

Factor $3$ out of $9 k^{2}$ .

$3 (k + 4) (3 (3 k^{2}) + 48)$

Step 3.14.1.2

Factor $3$ out of $48$ .

$3 (k + 4) (3 (3 k^{2}) + 3 (16))$

Step 3.14.1.3

Factor $3$ out of $3 (3 k^{2}) + 3 (16)$ .

$3 (k + 4) (3 (3 k^{2} + 16))$

Step 3.14.2

Remove unnecessary parentheses.

$3 (k + 4) \cdot 3 (3 k^{2} + 16)$

Step 3.15

Multiply $3$ by $3$ .

$9 (k + 4) (3 k^{2} + 16)$