Simplify (3a+2c)(9a^2-6ac+4c^2)

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

Step 1

Expand $(3 a + 2 c) (9 a^{2} - 6 a c + 4 c^{2})$ by multiplying each term in the first expression by each term in the second expression.

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

Step 2

Simplify terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

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

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

Move $a^{2}$ .

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

Multiply $a^{2}$ by $a$ .

Raise $a$ to the power of $1$ .

$3 \cdot 9 (a^{2} a^{1}) + 3 a (- 6 a c) + 3 a (4 c^{2}) + 2 c (9 a^{2}) + 2 c (- 6 a c) + 2 c (4 c^{2})$

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

$3 \cdot 9 a^{2 + 1} + 3 a (- 6 a c) + 3 a (4 c^{2}) + 2 c (9 a^{2}) + 2 c (- 6 a c) + 2 c (4 c^{2})$

Add $2$ and $1$ .

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

Multiply $3$ by $9$ .

$27 a^{3} + 3 a (- 6 a c) + 3 a (4 c^{2}) + 2 c (9 a^{2}) + 2 c (- 6 a c) + 2 c (4 c^{2})$

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

Move $a$ .

$27 a^{3} + 3 (a \cdot a) (- 6 c) + 3 a (4 c^{2}) + 2 c (9 a^{2}) + 2 c (- 6 a c) + 2 c (4 c^{2})$

Multiply $a$ by $a$ .

$27 a^{3} + 3 a^{2} (- 6 c) + 3 a (4 c^{2}) + 2 c (9 a^{2}) + 2 c (- 6 a c) + 2 c (4 c^{2})$

Rewrite using the commutative property of multiplication.

$27 a^{3} + 3 \cdot - 6 a^{2} c + 3 a (4 c^{2}) + 2 c (9 a^{2}) + 2 c (- 6 a c) + 2 c (4 c^{2})$

Multiply $3$ by $- 6$ .

$27 a^{3} - 18 a^{2} c + 3 a (4 c^{2}) + 2 c (9 a^{2}) + 2 c (- 6 a c) + 2 c (4 c^{2})$

Rewrite using the commutative property of multiplication.

$27 a^{3} - 18 a^{2} c + 3 \cdot 4 a c^{2} + 2 c (9 a^{2}) + 2 c (- 6 a c) + 2 c (4 c^{2})$

Multiply $3$ by $4$ .

$27 a^{3} - 18 a^{2} c + 12 a c^{2} + 2 c (9 a^{2}) + 2 c (- 6 a c) + 2 c (4 c^{2})$

Rewrite using the commutative property of multiplication.

$27 a^{3} - 18 a^{2} c + 12 a c^{2} + 2 \cdot 9 c a^{2} + 2 c (- 6 a c) + 2 c (4 c^{2})$

Multiply $2$ by $9$ .

$27 a^{3} - 18 a^{2} c + 12 a c^{2} + 18 c a^{2} + 2 c (- 6 a c) + 2 c (4 c^{2})$

Multiply $c$ by $c$ by adding the exponents.

Move $c$ .

$27 a^{3} - 18 a^{2} c + 12 a c^{2} + 18 c a^{2} + 2 (c \cdot c) (- 6 a) + 2 c (4 c^{2})$

Multiply $c$ by $c$ .

$27 a^{3} - 18 a^{2} c + 12 a c^{2} + 18 c a^{2} + 2 c^{2} (- 6 a) + 2 c (4 c^{2})$

Rewrite using the commutative property of multiplication.

$27 a^{3} - 18 a^{2} c + 12 a c^{2} + 18 c a^{2} + 2 \cdot - 6 c^{2} a + 2 c (4 c^{2})$

Multiply $2$ by $- 6$ .

$27 a^{3} - 18 a^{2} c + 12 a c^{2} + 18 c a^{2} - 12 c^{2} a + 2 c (4 c^{2})$

Rewrite using the commutative property of multiplication.

$27 a^{3} - 18 a^{2} c + 12 a c^{2} + 18 c a^{2} - 12 c^{2} a + 2 \cdot 4 c \cdot c^{2}$

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

Move $c^{2}$ .

$27 a^{3} - 18 a^{2} c + 12 a c^{2} + 18 c a^{2} - 12 c^{2} a + 2 \cdot 4 (c^{2} c)$

Multiply $c^{2}$ by $c$ .

Raise $c$ to the power of $1$ .

$27 a^{3} - 18 a^{2} c + 12 a c^{2} + 18 c a^{2} - 12 c^{2} a + 2 \cdot 4 (c^{2} c^{1})$

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

$27 a^{3} - 18 a^{2} c + 12 a c^{2} + 18 c a^{2} - 12 c^{2} a + 2 \cdot 4 c^{2 + 1}$

Add $2$ and $1$ .

$27 a^{3} - 18 a^{2} c + 12 a c^{2} + 18 c a^{2} - 12 c^{2} a + 2 \cdot 4 c^{3}$

Multiply $2$ by $4$ .

$27 a^{3} - 18 a^{2} c + 12 a c^{2} + 18 c a^{2} - 12 c^{2} a + 8 c^{3}$

Combine the opposite terms in $27 a^{3} - 18 a^{2} c + 12 a c^{2} + 18 c a^{2} - 12 c^{2} a + 8 c^{3}$ .

Reorder the factors in the terms $- 18 a^{2} c$ and $18 c a^{2}$ .

$27 a^{3} - 18 a^{2} c + 12 a c^{2} + 18 a^{2} c - 12 c^{2} a + 8 c^{3}$

Add $- 18 a^{2} c$ and $18 a^{2} c$ .

$27 a^{3} + 12 a c^{2} + 0 - 12 c^{2} a + 8 c^{3}$

Add $27 a^{3} + 12 a c^{2}$ and $0$ .

$27 a^{3} + 12 a c^{2} - 12 c^{2} a + 8 c^{3}$

Reorder the factors in the terms $12 a c^{2}$ and $- 12 c^{2} a$ .

$27 a^{3} + 12 c^{2} a - 12 c^{2} a + 8 c^{3}$

Subtract $12 c^{2} a$ from $12 c^{2} a$ .

$27 a^{3} + 0 + 8 c^{3}$

Add $27 a^{3}$ and $0$ .

$27 a^{3} + 8 c^{3}$

Simplify (3a+2c)(9a^2-6ac+4c^2)

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