Algebra Examples

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

Step 1

Rewrite $1$ as $1^{3}$ .

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

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 $a = 1$ and $b = a - 3 b$ .

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

Step 3

Simplify.

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Step 3.1

One to any power is one.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Rewrite $- 1 a$ as $- a$ .

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

Step 3.4

Multiply $- 3$ by $- 1$ .

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

Step 3.5

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

$(1 + a - 3 b) (1 - a + 3 b + (a - 3 b) (a - 3 b))$

Step 3.6

Expand $(a - 3 b) (a - 3 b)$ using the FOIL Method.

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Step 3.6.1

Apply the distributive property.

$(1 + a - 3 b) (1 - a + 3 b + a (a - 3 b) - 3 b (a - 3 b))$

Step 3.6.2

Apply the distributive property.

$(1 + a - 3 b) (1 - a + 3 b + a \cdot a + a (- 3 b) - 3 b (a - 3 b))$

Step 3.6.3

Apply the distributive property.

$(1 + a - 3 b) (1 - a + 3 b + a \cdot a + a (- 3 b) - 3 b a - 3 b (- 3 b))$

Step 3.7

Simplify and combine like terms.

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Step 3.7.1

Simplify each term.

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Step 3.7.1.1

Multiply $a$ by $a$ .

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

Step 3.7.1.2

Rewrite using the commutative property of multiplication.

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

Step 3.7.1.3

Rewrite using the commutative property of multiplication.

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

Step 3.7.1.4

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

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Step 3.7.1.4.1

Move $b$ .

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

Step 3.7.1.4.2

Multiply $b$ by $b$ .

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

Step 3.7.1.5

Multiply $- 3$ by $- 3$ .

$(1 + a - 3 b) (1 - a + 3 b + a^{2} - 3 a b - 3 b a + 9 b^{2})$

Step 3.7.2

Subtract $3 b a$ from $- 3 a b$ .

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Step 3.7.2.1

Move $b$ .

$(1 + a - 3 b) (1 - a + 3 b + a^{2} - 3 a b - 3 a b + 9 b^{2})$

Step 3.7.2.2

Subtract $3 a b$ from $- 3 a b$ .

$(1 + a - 3 b) (1 - a + 3 b + a^{2} - 6 a b + 9 b^{2})$