Expand Using the Binomial Theorem (4c+2h)^3

${(4 c + 2 h)}^{3}$

Use the binomial expansion theorem to find each term. The binomial theorem states ${(a + b)}^{n} = \sum_{k = 0}^{n} n C k \cdot (a^{n - k} b^{k})$ .

$\sum_{k = 0}^{3} \frac{3!}{(3 - k)! k!} \cdot {(4 c)}^{3 - k} \cdot {(2 h)}^{k}$

Expand the summation.

$\frac{3!}{(3 - 0)! 0!} \cdot {(4 c)}^{3 - 0} \cdot {(2 h)}^{0} + \frac{3!}{(3 - 1)! 1!} \cdot {(4 c)}^{3 - 1} \cdot {(2 h)}^{1} + \frac{3!}{(3 - 2)! 2!} \cdot {(4 c)}^{3 - 2} \cdot {(2 h)}^{2} + \frac{3!}{(3 - 3)! 3!} \cdot {(4 c)}^{3 - 3} \cdot {(2 h)}^{3}$

Simplify the exponents for each term of the expansion.

$1 \cdot {(4 c)}^{3} \cdot {(2 h)}^{0} + 3 \cdot {(4 c)}^{2} \cdot {(2 h)}^{1} + 3 \cdot {(4 c)}^{1} \cdot {(2 h)}^{2} + 1 \cdot {(4 c)}^{0} \cdot {(2 h)}^{3}$

Simplify each term.

Multiply ${(4 c)}^{3}$ by $1$ .

${(4 c)}^{3} \cdot {(2 h)}^{0} + 3 \cdot {(4 c)}^{2} \cdot {(2 h)}^{1} + 3 \cdot {(4 c)}^{1} \cdot {(2 h)}^{2} + 1 \cdot {(4 c)}^{0} \cdot {(2 h)}^{3}$

Apply the product rule to $4 c$ .

$4^{3} c^{3} \cdot {(2 h)}^{0} + 3 \cdot {(4 c)}^{2} \cdot {(2 h)}^{1} + 3 \cdot {(4 c)}^{1} \cdot {(2 h)}^{2} + 1 \cdot {(4 c)}^{0} \cdot {(2 h)}^{3}$

Raise $4$ to the power of $3$ .

$64 c^{3} \cdot {(2 h)}^{0} + 3 \cdot {(4 c)}^{2} \cdot {(2 h)}^{1} + 3 \cdot {(4 c)}^{1} \cdot {(2 h)}^{2} + 1 \cdot {(4 c)}^{0} \cdot {(2 h)}^{3}$

Apply the product rule to $2 h$ .

$64 c^{3} \cdot (2^{0} h^{0}) + 3 \cdot {(4 c)}^{2} \cdot {(2 h)}^{1} + 3 \cdot {(4 c)}^{1} \cdot {(2 h)}^{2} + 1 \cdot {(4 c)}^{0} \cdot {(2 h)}^{3}$

Rewrite using the commutative property of multiplication.

$64 \cdot 2^{0} c^{3} h^{0} + 3 \cdot {(4 c)}^{2} \cdot {(2 h)}^{1} + 3 \cdot {(4 c)}^{1} \cdot {(2 h)}^{2} + 1 \cdot {(4 c)}^{0} \cdot {(2 h)}^{3}$

Anything raised to $0$ is $1$ .

$64 \cdot 1 c^{3} h^{0} + 3 \cdot {(4 c)}^{2} \cdot {(2 h)}^{1} + 3 \cdot {(4 c)}^{1} \cdot {(2 h)}^{2} + 1 \cdot {(4 c)}^{0} \cdot {(2 h)}^{3}$

Multiply $64$ by $1$ .

$64 c^{3} h^{0} + 3 \cdot {(4 c)}^{2} \cdot {(2 h)}^{1} + 3 \cdot {(4 c)}^{1} \cdot {(2 h)}^{2} + 1 \cdot {(4 c)}^{0} \cdot {(2 h)}^{3}$

Anything raised to $0$ is $1$ .

$64 c^{3} \cdot 1 + 3 \cdot {(4 c)}^{2} \cdot {(2 h)}^{1} + 3 \cdot {(4 c)}^{1} \cdot {(2 h)}^{2} + 1 \cdot {(4 c)}^{0} \cdot {(2 h)}^{3}$

Multiply $64$ by $1$ .

$64 c^{3} + 3 \cdot {(4 c)}^{2} \cdot {(2 h)}^{1} + 3 \cdot {(4 c)}^{1} \cdot {(2 h)}^{2} + 1 \cdot {(4 c)}^{0} \cdot {(2 h)}^{3}$

Apply the product rule to $4 c$ .

$64 c^{3} + 3 \cdot (4^{2} c^{2}) \cdot {(2 h)}^{1} + 3 \cdot {(4 c)}^{1} \cdot {(2 h)}^{2} + 1 \cdot {(4 c)}^{0} \cdot {(2 h)}^{3}$

Raise $4$ to the power of $2$ .

$64 c^{3} + 3 \cdot (16 c^{2}) \cdot {(2 h)}^{1} + 3 \cdot {(4 c)}^{1} \cdot {(2 h)}^{2} + 1 \cdot {(4 c)}^{0} \cdot {(2 h)}^{3}$

Multiply $16$ by $3$ .

$64 c^{3} + 48 c^{2} \cdot {(2 h)}^{1} + 3 \cdot {(4 c)}^{1} \cdot {(2 h)}^{2} + 1 \cdot {(4 c)}^{0} \cdot {(2 h)}^{3}$

Simplify.

$64 c^{3} + 48 c^{2} \cdot (2 h) + 3 \cdot {(4 c)}^{1} \cdot {(2 h)}^{2} + 1 \cdot {(4 c)}^{0} \cdot {(2 h)}^{3}$

Rewrite using the commutative property of multiplication.

$64 c^{3} + 48 \cdot 2 c^{2} h + 3 \cdot {(4 c)}^{1} \cdot {(2 h)}^{2} + 1 \cdot {(4 c)}^{0} \cdot {(2 h)}^{3}$

Multiply $48$ by $2$ .

$64 c^{3} + 96 c^{2} h + 3 \cdot {(4 c)}^{1} \cdot {(2 h)}^{2} + 1 \cdot {(4 c)}^{0} \cdot {(2 h)}^{3}$

Simplify.

$64 c^{3} + 96 c^{2} h + 3 \cdot (4 c) \cdot {(2 h)}^{2} + 1 \cdot {(4 c)}^{0} \cdot {(2 h)}^{3}$

Multiply $4$ by $3$ .

$64 c^{3} + 96 c^{2} h + 12 c \cdot {(2 h)}^{2} + 1 \cdot {(4 c)}^{0} \cdot {(2 h)}^{3}$

Apply the product rule to $2 h$ .

$64 c^{3} + 96 c^{2} h + 12 c \cdot (2^{2} h^{2}) + 1 \cdot {(4 c)}^{0} \cdot {(2 h)}^{3}$

Rewrite using the commutative property of multiplication.

$64 c^{3} + 96 c^{2} h + 12 \cdot 2^{2} c h^{2} + 1 \cdot {(4 c)}^{0} \cdot {(2 h)}^{3}$

Raise $2$ to the power of $2$ .

$64 c^{3} + 96 c^{2} h + 12 \cdot 4 c h^{2} + 1 \cdot {(4 c)}^{0} \cdot {(2 h)}^{3}$

Multiply $12$ by $4$ .

$64 c^{3} + 96 c^{2} h + 48 c h^{2} + 1 \cdot {(4 c)}^{0} \cdot {(2 h)}^{3}$

Multiply ${(4 c)}^{0}$ by $1$ .

$64 c^{3} + 96 c^{2} h + 48 c h^{2} + {(4 c)}^{0} \cdot {(2 h)}^{3}$

Apply the product rule to $4 c$ .

$64 c^{3} + 96 c^{2} h + 48 c h^{2} + 4^{0} c^{0} \cdot {(2 h)}^{3}$

Anything raised to $0$ is $1$ .

$64 c^{3} + 96 c^{2} h + 48 c h^{2} + 1 c^{0} \cdot {(2 h)}^{3}$

Multiply $c^{0}$ by $1$ .

$64 c^{3} + 96 c^{2} h + 48 c h^{2} + c^{0} \cdot {(2 h)}^{3}$

Anything raised to $0$ is $1$ .

$64 c^{3} + 96 c^{2} h + 48 c h^{2} + 1 \cdot {(2 h)}^{3}$

Multiply ${(2 h)}^{3}$ by $1$ .

$64 c^{3} + 96 c^{2} h + 48 c h^{2} + {(2 h)}^{3}$

Apply the product rule to $2 h$ .

$64 c^{3} + 96 c^{2} h + 48 c h^{2} + 2^{3} h^{3}$

Raise $2$ to the power of $3$ .

$64 c^{3} + 96 c^{2} h + 48 c h^{2} + 8 h^{3}$

Expand Using the Binomial Theorem (4c+2h)^3

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