Pre-Algebra Examples

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

Step 1

Rewrite ${(a - b - 1)}^{2}$ as $(a - b - 1) (a - b - 1)$ .

$(a - b - 1) (a - b - 1)$

Step 2

Expand $(a - b - 1) (a - b - 1)$ by multiplying each term in the first expression by each term in the second expression.

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

Step 3

Simplify each term.

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Step 3.1

Multiply $a$ by $a$ .

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

Step 3.2

Rewrite using the commutative property of multiplication.

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

Step 3.3

Move $- 1$ to the left of $a$ .

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

Step 3.4

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

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

Step 3.5

Rewrite using the commutative property of multiplication.

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

Step 3.6

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

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Step 3.6.1

Move $b$ .

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

Step 3.6.2

Multiply $b$ by $b$ .

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

Step 3.7

Multiply $- 1$ by $- 1$ .

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

Step 3.8

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

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

Step 3.9

Multiply $- b \cdot - 1$ .

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Step 3.9.1

Multiply $- 1$ by $- 1$ .

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

Step 3.9.2

Multiply $b$ by $1$ .

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

Step 3.10

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

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

Step 3.11

Multiply $- 1 (- b)$ .

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Step 3.11.1

Multiply $- 1$ by $- 1$ .

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

Step 3.11.2

Multiply $b$ by $1$ .

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

Step 3.12

Multiply $- 1$ by $- 1$ .

$a^{2} - a b - a - b a + b^{2} + b - a + b + 1$

Step 4

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

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Step 4.1

Move $b$ .

$a^{2} - a - a b - 1 a b + b^{2} + b - a + b + 1$

Step 4.2

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

$a^{2} - a - 2 a b + b^{2} + b - a + b + 1$

Step 5

Subtract $a$ from $- a$ .

$a^{2} - 2 a - 2 a b + b^{2} + b + b + 1$

Step 6

Add $b$ and $b$ .

$a^{2} - 2 a - 2 a b + b^{2} + 2 b + 1$

Step 7

Move $- 2 a$ .

$a^{2} - 2 a b + b^{2} - 2 a + 2 b + 1$