Simplify (a^3-b^3)^2

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

Step 1

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

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

Step 2

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

Simplify each term.

Multiply $a^{3}$ by $a^{3}$ by adding the exponents.

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Add $3$ and $3$ .

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

Rewrite using the commutative property of multiplication.

$a^{6} - a^{3} b^{3} - b^{3} a^{3} - b^{3} (- b^{3})$

Rewrite using the commutative property of multiplication.

$a^{6} - a^{3} b^{3} - b^{3} a^{3} - 1 \cdot - 1 b^{3} b^{3}$

Multiply $b^{3}$ by $b^{3}$ by adding the exponents.

Move $b^{3}$ .

$a^{6} - a^{3} b^{3} - b^{3} a^{3} - 1 \cdot - 1 (b^{3} b^{3})$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$a^{6} - a^{3} b^{3} - b^{3} a^{3} - 1 \cdot - 1 b^{3 + 3}$

Add $3$ and $3$ .

$a^{6} - a^{3} b^{3} - b^{3} a^{3} - 1 \cdot - 1 b^{6}$

Multiply $- 1$ by $- 1$ .

$a^{6} - a^{3} b^{3} - b^{3} a^{3} + 1 b^{6}$

Multiply $b^{6}$ by $1$ .

$a^{6} - a^{3} b^{3} - b^{3} a^{3} + b^{6}$

Subtract $b^{3} a^{3}$ from $- a^{3} b^{3}$ .

Move $b^{3}$ .

$a^{6} - a^{3} b^{3} - 1 a^{3} b^{3} + b^{6}$

Subtract $a^{3} b^{3}$ from $- a^{3} b^{3}$ .

$a^{6} - 2 a^{3} b^{3} + b^{6}$