Expand the Trigonometric Expression (4a^2-3b^2)(16a^4+12a^2b^2+9b^4)

$(4 a^{2} - 3 b^{2}) (16 a^{4} + 12 a^{2} b^{2} + 9 b^{4})$

Expand $(4 a^{2} - 3 b^{2}) (16 a^{4} + 12 a^{2} b^{2} + 9 b^{4})$ by multiplying each term in the first expression by each term in the second expression.

$4 a^{2} (16 a^{4}) + 4 a^{2} (12 a^{2} b^{2}) + 4 a^{2} (9 b^{4}) - 3 b^{2} (16 a^{4}) - 3 b^{2} (12 a^{2} b^{2}) - 3 b^{2} (9 b^{4})$

Simplify terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$4 \cdot 16 a^{2} a^{4} + 4 a^{2} (12 a^{2} b^{2}) + 4 a^{2} (9 b^{4}) - 3 b^{2} (16 a^{4}) - 3 b^{2} (12 a^{2} b^{2}) - 3 b^{2} (9 b^{4})$

Multiply $a^{2}$ by $a^{4}$ by adding the exponents.

Move $a^{4}$ .

$4 \cdot 16 (a^{4} a^{2}) + 4 a^{2} (12 a^{2} b^{2}) + 4 a^{2} (9 b^{4}) - 3 b^{2} (16 a^{4}) - 3 b^{2} (12 a^{2} b^{2}) - 3 b^{2} (9 b^{4})$

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

$4 \cdot 16 a^{4 + 2} + 4 a^{2} (12 a^{2} b^{2}) + 4 a^{2} (9 b^{4}) - 3 b^{2} (16 a^{4}) - 3 b^{2} (12 a^{2} b^{2}) - 3 b^{2} (9 b^{4})$

Add $4$ and $2$ .

$4 \cdot 16 a^{6} + 4 a^{2} (12 a^{2} b^{2}) + 4 a^{2} (9 b^{4}) - 3 b^{2} (16 a^{4}) - 3 b^{2} (12 a^{2} b^{2}) - 3 b^{2} (9 b^{4})$

Multiply $4$ by $16$ .

$64 a^{6} + 4 a^{2} (12 a^{2} b^{2}) + 4 a^{2} (9 b^{4}) - 3 b^{2} (16 a^{4}) - 3 b^{2} (12 a^{2} b^{2}) - 3 b^{2} (9 b^{4})$

Multiply $a^{2}$ by $a^{2}$ by adding the exponents.

Move $a^{2}$ .

$64 a^{6} + 4 (a^{2} a^{2}) (12 b^{2}) + 4 a^{2} (9 b^{4}) - 3 b^{2} (16 a^{4}) - 3 b^{2} (12 a^{2} b^{2}) - 3 b^{2} (9 b^{4})$

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

$64 a^{6} + 4 a^{2 + 2} (12 b^{2}) + 4 a^{2} (9 b^{4}) - 3 b^{2} (16 a^{4}) - 3 b^{2} (12 a^{2} b^{2}) - 3 b^{2} (9 b^{4})$

Add $2$ and $2$ .

$64 a^{6} + 4 a^{4} (12 b^{2}) + 4 a^{2} (9 b^{4}) - 3 b^{2} (16 a^{4}) - 3 b^{2} (12 a^{2} b^{2}) - 3 b^{2} (9 b^{4})$

Rewrite using the commutative property of multiplication.

$64 a^{6} + 4 \cdot 12 a^{4} b^{2} + 4 a^{2} (9 b^{4}) - 3 b^{2} (16 a^{4}) - 3 b^{2} (12 a^{2} b^{2}) - 3 b^{2} (9 b^{4})$

Multiply $4$ by $12$ .

$64 a^{6} + 48 a^{4} b^{2} + 4 a^{2} (9 b^{4}) - 3 b^{2} (16 a^{4}) - 3 b^{2} (12 a^{2} b^{2}) - 3 b^{2} (9 b^{4})$

Rewrite using the commutative property of multiplication.

$64 a^{6} + 48 a^{4} b^{2} + 4 \cdot 9 a^{2} b^{4} - 3 b^{2} (16 a^{4}) - 3 b^{2} (12 a^{2} b^{2}) - 3 b^{2} (9 b^{4})$

Multiply $4$ by $9$ .

$64 a^{6} + 48 a^{4} b^{2} + 36 a^{2} b^{4} - 3 b^{2} (16 a^{4}) - 3 b^{2} (12 a^{2} b^{2}) - 3 b^{2} (9 b^{4})$

Rewrite using the commutative property of multiplication.

$64 a^{6} + 48 a^{4} b^{2} + 36 a^{2} b^{4} - 3 \cdot 16 b^{2} a^{4} - 3 b^{2} (12 a^{2} b^{2}) - 3 b^{2} (9 b^{4})$

Multiply $- 3$ by $16$ .

$64 a^{6} + 48 a^{4} b^{2} + 36 a^{2} b^{4} - 48 b^{2} a^{4} - 3 b^{2} (12 a^{2} b^{2}) - 3 b^{2} (9 b^{4})$

Multiply $b^{2}$ by $b^{2}$ by adding the exponents.

Move $b^{2}$ .

$64 a^{6} + 48 a^{4} b^{2} + 36 a^{2} b^{4} - 48 b^{2} a^{4} - 3 (b^{2} b^{2}) (12 a^{2}) - 3 b^{2} (9 b^{4})$

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

$64 a^{6} + 48 a^{4} b^{2} + 36 a^{2} b^{4} - 48 b^{2} a^{4} - 3 b^{2 + 2} (12 a^{2}) - 3 b^{2} (9 b^{4})$

Add $2$ and $2$ .

$64 a^{6} + 48 a^{4} b^{2} + 36 a^{2} b^{4} - 48 b^{2} a^{4} - 3 b^{4} (12 a^{2}) - 3 b^{2} (9 b^{4})$

Rewrite using the commutative property of multiplication.

$64 a^{6} + 48 a^{4} b^{2} + 36 a^{2} b^{4} - 48 b^{2} a^{4} - 3 \cdot 12 b^{4} a^{2} - 3 b^{2} (9 b^{4})$

Multiply $- 3$ by $12$ .

$64 a^{6} + 48 a^{4} b^{2} + 36 a^{2} b^{4} - 48 b^{2} a^{4} - 36 b^{4} a^{2} - 3 b^{2} (9 b^{4})$

Rewrite using the commutative property of multiplication.

$64 a^{6} + 48 a^{4} b^{2} + 36 a^{2} b^{4} - 48 b^{2} a^{4} - 36 b^{4} a^{2} - 3 \cdot 9 b^{2} b^{4}$

Multiply $b^{2}$ by $b^{4}$ by adding the exponents.

Move $b^{4}$ .

$64 a^{6} + 48 a^{4} b^{2} + 36 a^{2} b^{4} - 48 b^{2} a^{4} - 36 b^{4} a^{2} - 3 \cdot 9 (b^{4} b^{2})$

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

$64 a^{6} + 48 a^{4} b^{2} + 36 a^{2} b^{4} - 48 b^{2} a^{4} - 36 b^{4} a^{2} - 3 \cdot 9 b^{4 + 2}$

Add $4$ and $2$ .

$64 a^{6} + 48 a^{4} b^{2} + 36 a^{2} b^{4} - 48 b^{2} a^{4} - 36 b^{4} a^{2} - 3 \cdot 9 b^{6}$

Multiply $- 3$ by $9$ .

$64 a^{6} + 48 a^{4} b^{2} + 36 a^{2} b^{4} - 48 b^{2} a^{4} - 36 b^{4} a^{2} - 27 b^{6}$

Combine the opposite terms in $64 a^{6} + 48 a^{4} b^{2} + 36 a^{2} b^{4} - 48 b^{2} a^{4} - 36 b^{4} a^{2} - 27 b^{6}$ .

Reorder the factors in the terms $48 a^{4} b^{2}$ and $- 48 b^{2} a^{4}$ .

$64 a^{6} + 48 a^{4} b^{2} + 36 a^{2} b^{4} - 48 a^{4} b^{2} - 36 b^{4} a^{2} - 27 b^{6}$

Subtract $48 a^{4} b^{2}$ from $48 a^{4} b^{2}$ .

$64 a^{6} + 36 a^{2} b^{4} + 0 - 36 b^{4} a^{2} - 27 b^{6}$

Add $64 a^{6} + 36 a^{2} b^{4}$ and $0$ .

$64 a^{6} + 36 a^{2} b^{4} - 36 b^{4} a^{2} - 27 b^{6}$

Reorder the factors in the terms $36 a^{2} b^{4}$ and $- 36 b^{4} a^{2}$ .

$64 a^{6} + 36 b^{4} a^{2} - 36 b^{4} a^{2} - 27 b^{6}$

Subtract $36 b^{4} a^{2}$ from $36 b^{4} a^{2}$ .

$64 a^{6} + 0 - 27 b^{6}$

Add $64 a^{6}$ and $0$ .

$64 a^{6} - 27 b^{6}$

Expand the Trigonometric Expression (4a^2-3b^2)(16a^4+12a^2b^2+9b^4)

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