Simplify (ab-1)^2

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

Step 1

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

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

Step 2

Expand $(a b - 1) (a b - 1)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

Simplify each term.

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

Move $a$ .

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

Multiply $a$ by $a$ .

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

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

Move $b$ .

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

Multiply $b$ by $b$ .

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

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

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

Rewrite $- 1 (a b)$ as $- (a b)$ .

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

Rewrite $- 1 (a b)$ as $- (a b)$ .

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

Multiply $- 1$ by $- 1$ .

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

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

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