Find the Product (2a^6-6a^3)^2

${(2 a^{6} - 6 a^{3})}^{2}$

Step 1

Rewrite ${(2 a^{6} - 6 a^{3})}^{2}$ as $(2 a^{6} - 6 a^{3}) (2 a^{6} - 6 a^{3})$ .

$(2 a^{6} - 6 a^{3}) (2 a^{6} - 6 a^{3})$

Step 2

Expand $(2 a^{6} - 6 a^{3}) (2 a^{6} - 6 a^{3})$ using the FOIL Method.

Apply the distributive property.

$2 a^{6} (2 a^{6} - 6 a^{3}) - 6 a^{3} (2 a^{6} - 6 a^{3})$

Apply the distributive property.

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

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$2 \cdot 2 a^{6} a^{6} + 2 a^{6} (- 6 a^{3}) - 6 a^{3} (2 a^{6}) - 6 a^{3} (- 6 a^{3})$

Multiply $a^{6}$ by $a^{6}$ by adding the exponents.

Move $a^{6}$ .

$2 \cdot 2 (a^{6} a^{6}) + 2 a^{6} (- 6 a^{3}) - 6 a^{3} (2 a^{6}) - 6 a^{3} (- 6 a^{3})$

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

$2 \cdot 2 a^{6 + 6} + 2 a^{6} (- 6 a^{3}) - 6 a^{3} (2 a^{6}) - 6 a^{3} (- 6 a^{3})$

Add $6$ and $6$ .

$2 \cdot 2 a^{12} + 2 a^{6} (- 6 a^{3}) - 6 a^{3} (2 a^{6}) - 6 a^{3} (- 6 a^{3})$

Multiply $2$ by $2$ .

$4 a^{12} + 2 a^{6} (- 6 a^{3}) - 6 a^{3} (2 a^{6}) - 6 a^{3} (- 6 a^{3})$

Rewrite using the commutative property of multiplication.

$4 a^{12} + 2 \cdot - 6 a^{6} a^{3} - 6 a^{3} (2 a^{6}) - 6 a^{3} (- 6 a^{3})$

Multiply $a^{6}$ by $a^{3}$ by adding the exponents.

Move $a^{3}$ .

$4 a^{12} + 2 \cdot - 6 (a^{3} a^{6}) - 6 a^{3} (2 a^{6}) - 6 a^{3} (- 6 a^{3})$

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

$4 a^{12} + 2 \cdot - 6 a^{3 + 6} - 6 a^{3} (2 a^{6}) - 6 a^{3} (- 6 a^{3})$

Add $3$ and $6$ .

$4 a^{12} + 2 \cdot - 6 a^{9} - 6 a^{3} (2 a^{6}) - 6 a^{3} (- 6 a^{3})$

Multiply $2$ by $- 6$ .

$4 a^{12} - 12 a^{9} - 6 a^{3} (2 a^{6}) - 6 a^{3} (- 6 a^{3})$

Rewrite using the commutative property of multiplication.

$4 a^{12} - 12 a^{9} - 6 \cdot 2 a^{3} a^{6} - 6 a^{3} (- 6 a^{3})$

Multiply $a^{3}$ by $a^{6}$ by adding the exponents.

Move $a^{6}$ .

$4 a^{12} - 12 a^{9} - 6 \cdot 2 (a^{6} a^{3}) - 6 a^{3} (- 6 a^{3})$

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

$4 a^{12} - 12 a^{9} - 6 \cdot 2 a^{6 + 3} - 6 a^{3} (- 6 a^{3})$

Add $6$ and $3$ .

$4 a^{12} - 12 a^{9} - 6 \cdot 2 a^{9} - 6 a^{3} (- 6 a^{3})$

Multiply $- 6$ by $2$ .

$4 a^{12} - 12 a^{9} - 12 a^{9} - 6 a^{3} (- 6 a^{3})$

Rewrite using the commutative property of multiplication.

$4 a^{12} - 12 a^{9} - 12 a^{9} - 6 \cdot - 6 a^{3} a^{3}$

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

Move $a^{3}$ .

$4 a^{12} - 12 a^{9} - 12 a^{9} - 6 \cdot - 6 (a^{3} a^{3})$

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

$4 a^{12} - 12 a^{9} - 12 a^{9} - 6 \cdot - 6 a^{3 + 3}$

Add $3$ and $3$ .

$4 a^{12} - 12 a^{9} - 12 a^{9} - 6 \cdot - 6 a^{6}$

Multiply $- 6$ by $- 6$ .

$4 a^{12} - 12 a^{9} - 12 a^{9} + 36 a^{6}$

Subtract $12 a^{9}$ from $- 12 a^{9}$ .

$4 a^{12} - 24 a^{9} + 36 a^{6}$

Find the Product (2a^6-6a^3)^2

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