Basic Math Examples

${(a - b)}^{3} + 27$

Step 1

Rewrite $27$ as $3^{3}$ .

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

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 $a = a - b$ and $b = 3$ .

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

Step 3

Simplify.

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

Apply the distributive property.

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

Step 3.2.2

Apply the distributive property.

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

Step 3.2.3

Apply the distributive property.

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

Step 3.3

Simplify and combine like terms.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Multiply $a$ by $a$ .

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

Step 3.3.1.2

Rewrite using the commutative property of multiplication.

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

Step 3.3.1.3

Rewrite using the commutative property of multiplication.

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

Step 3.3.1.4

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

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Step 3.3.1.4.1

Move $b$ .

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

Step 3.3.1.4.2

Multiply $b$ by $b$ .

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

Step 3.3.1.5

Multiply $- 1$ by $- 1$ .

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

Step 3.3.1.6

Multiply $b^{2}$ by $1$ .

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

Step 3.3.2

Subtract $b a$ from $- a b$ .

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Step 3.3.2.1

Move $b$ .

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

Step 3.3.2.2

Subtract $a b$ from $- a b$ .

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

Step 3.4

Apply the distributive property.

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

Step 3.5

Multiply $- - b$ .

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Step 3.5.1

Multiply $- 1$ by $- 1$ .

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

Step 3.5.2

Multiply $b$ by $1$ .

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

Step 3.6

Apply the distributive property.

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

Step 3.7

Multiply $3$ by $- 1$ .

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

Step 3.8

Move $3$ to the left of $b$ .

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

Step 3.9

Raise $3$ to the power of $2$ .

$(a - b + 3) (a^{2} - 2 a b + b^{2} - 3 a + 3 b + 9)$