Expand Using the Binomial Theorem (12x-yz)^3

${(12 x - y z)}^{3}$

Step 1

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

$\sum_{k = 0}^{3} \frac{3!}{(3 - k)! k!} \cdot {(12 x)}^{3 - k} \cdot {(- y z)}^{k}$

Step 2

Expand the summation.

$\frac{3!}{(3 - 0)! 0!} \cdot {(12 x)}^{3 - 0} \cdot {(- y z)}^{0} + \frac{3!}{(3 - 1)! 1!} \cdot {(12 x)}^{3 - 1} \cdot {(- y z)}^{1} + \frac{3!}{(3 - 2)! 2!} \cdot {(12 x)}^{3 - 2} \cdot {(- y z)}^{2} + \frac{3!}{(3 - 3)! 3!} \cdot {(12 x)}^{3 - 3} \cdot {(- y z)}^{3}$

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(12 x)}^{3} \cdot {(- y z)}^{0} + 3 \cdot {(12 x)}^{2} \cdot {(- y z)}^{1} + 3 \cdot {(12 x)}^{1} \cdot {(- y z)}^{2} + 1 \cdot {(12 x)}^{0} \cdot {(- y z)}^{3}$

Step 4

Simplify each term.

Multiply ${(12 x)}^{3}$ by $1$ .

${(12 x)}^{3} \cdot {(- y z)}^{0} + 3 \cdot {(12 x)}^{2} \cdot {(- y z)}^{1} + 3 \cdot {(12 x)}^{1} \cdot {(- y z)}^{2} + 1 \cdot {(12 x)}^{0} \cdot {(- y z)}^{3}$

Apply the product rule to $12 x$ .

$12^{3} x^{3} \cdot {(- y z)}^{0} + 3 \cdot {(12 x)}^{2} \cdot {(- y z)}^{1} + 3 \cdot {(12 x)}^{1} \cdot {(- y z)}^{2} + 1 \cdot {(12 x)}^{0} \cdot {(- y z)}^{3}$

Raise $12$ to the power of $3$ .

$1728 x^{3} \cdot {(- y z)}^{0} + 3 \cdot {(12 x)}^{2} \cdot {(- y z)}^{1} + 3 \cdot {(12 x)}^{1} \cdot {(- y z)}^{2} + 1 \cdot {(12 x)}^{0} \cdot {(- y z)}^{3}$

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $- y z$ .

$1728 x^{3} \cdot ({(- y)}^{0} z^{0}) + 3 \cdot {(12 x)}^{2} \cdot {(- y z)}^{1} + 3 \cdot {(12 x)}^{1} \cdot {(- y z)}^{2} + 1 \cdot {(12 x)}^{0} \cdot {(- y z)}^{3}$

Apply the product rule to $- y$ .

$1728 x^{3} \cdot ({(- 1)}^{0} y^{0} z^{0}) + 3 \cdot {(12 x)}^{2} \cdot {(- y z)}^{1} + 3 \cdot {(12 x)}^{1} \cdot {(- y z)}^{2} + 1 \cdot {(12 x)}^{0} \cdot {(- y z)}^{3}$

Anything raised to $0$ is $1$ .

$1728 x^{3} \cdot (1 y^{0} z^{0}) + 3 \cdot {(12 x)}^{2} \cdot {(- y z)}^{1} + 3 \cdot {(12 x)}^{1} \cdot {(- y z)}^{2} + 1 \cdot {(12 x)}^{0} \cdot {(- y z)}^{3}$

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

$1728 x^{3} \cdot (y^{0} z^{0}) + 3 \cdot {(12 x)}^{2} \cdot {(- y z)}^{1} + 3 \cdot {(12 x)}^{1} \cdot {(- y z)}^{2} + 1 \cdot {(12 x)}^{0} \cdot {(- y z)}^{3}$

Anything raised to $0$ is $1$ .

$1728 x^{3} \cdot (1 z^{0}) + 3 \cdot {(12 x)}^{2} \cdot {(- y z)}^{1} + 3 \cdot {(12 x)}^{1} \cdot {(- y z)}^{2} + 1 \cdot {(12 x)}^{0} \cdot {(- y z)}^{3}$

Rewrite using the commutative property of multiplication.

$1728 \cdot 1 x^{3} z^{0} + 3 \cdot {(12 x)}^{2} \cdot {(- y z)}^{1} + 3 \cdot {(12 x)}^{1} \cdot {(- y z)}^{2} + 1 \cdot {(12 x)}^{0} \cdot {(- y z)}^{3}$

Multiply $1728$ by $1$ .

$1728 x^{3} z^{0} + 3 \cdot {(12 x)}^{2} \cdot {(- y z)}^{1} + 3 \cdot {(12 x)}^{1} \cdot {(- y z)}^{2} + 1 \cdot {(12 x)}^{0} \cdot {(- y z)}^{3}$

Anything raised to $0$ is $1$ .

$1728 x^{3} \cdot 1 + 3 \cdot {(12 x)}^{2} \cdot {(- y z)}^{1} + 3 \cdot {(12 x)}^{1} \cdot {(- y z)}^{2} + 1 \cdot {(12 x)}^{0} \cdot {(- y z)}^{3}$

Multiply $1728$ by $1$ .

$1728 x^{3} + 3 \cdot {(12 x)}^{2} \cdot {(- y z)}^{1} + 3 \cdot {(12 x)}^{1} \cdot {(- y z)}^{2} + 1 \cdot {(12 x)}^{0} \cdot {(- y z)}^{3}$

Apply the product rule to $12 x$ .

$1728 x^{3} + 3 \cdot (12^{2} x^{2}) \cdot {(- y z)}^{1} + 3 \cdot {(12 x)}^{1} \cdot {(- y z)}^{2} + 1 \cdot {(12 x)}^{0} \cdot {(- y z)}^{3}$

Raise $12$ to the power of $2$ .

$1728 x^{3} + 3 \cdot (144 x^{2}) \cdot {(- y z)}^{1} + 3 \cdot {(12 x)}^{1} \cdot {(- y z)}^{2} + 1 \cdot {(12 x)}^{0} \cdot {(- y z)}^{3}$

Multiply $144$ by $3$ .

$1728 x^{3} + 432 x^{2} \cdot {(- y z)}^{1} + 3 \cdot {(12 x)}^{1} \cdot {(- y z)}^{2} + 1 \cdot {(12 x)}^{0} \cdot {(- y z)}^{3}$

Simplify.

$1728 x^{3} + 432 x^{2} \cdot (- y z) + 3 \cdot {(12 x)}^{1} \cdot {(- y z)}^{2} + 1 \cdot {(12 x)}^{0} \cdot {(- y z)}^{3}$

Multiply $- 1$ by $432$ .

$1728 x^{3} - 432 x^{2} (y z) + 3 \cdot {(12 x)}^{1} \cdot {(- y z)}^{2} + 1 \cdot {(12 x)}^{0} \cdot {(- y z)}^{3}$

Simplify.

$1728 x^{3} - 432 x^{2} y z + 3 \cdot (12 x) \cdot {(- y z)}^{2} + 1 \cdot {(12 x)}^{0} \cdot {(- y z)}^{3}$

Multiply $12$ by $3$ .

$1728 x^{3} - 432 x^{2} y z + 36 x \cdot {(- y z)}^{2} + 1 \cdot {(12 x)}^{0} \cdot {(- y z)}^{3}$

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $- y z$ .

$1728 x^{3} - 432 x^{2} y z + 36 x \cdot ({(- y)}^{2} z^{2}) + 1 \cdot {(12 x)}^{0} \cdot {(- y z)}^{3}$

Apply the product rule to $- y$ .

$1728 x^{3} - 432 x^{2} y z + 36 x \cdot ({(- 1)}^{2} y^{2} z^{2}) + 1 \cdot {(12 x)}^{0} \cdot {(- y z)}^{3}$

Raise $- 1$ to the power of $2$ .

$1728 x^{3} - 432 x^{2} y z + 36 x \cdot (1 y^{2} z^{2}) + 1 \cdot {(12 x)}^{0} \cdot {(- y z)}^{3}$

Multiply $y^{2}$ by $1$ .

$1728 x^{3} - 432 x^{2} y z + 36 x \cdot (y^{2} z^{2}) + 1 \cdot {(12 x)}^{0} \cdot {(- y z)}^{3}$

Multiply ${(12 x)}^{0}$ by $1$ .

$1728 x^{3} - 432 x^{2} y z + 36 x y^{2} z^{2} + {(12 x)}^{0} \cdot {(- y z)}^{3}$

Apply the product rule to $12 x$ .

$1728 x^{3} - 432 x^{2} y z + 36 x y^{2} z^{2} + 12^{0} x^{0} \cdot {(- y z)}^{3}$

Anything raised to $0$ is $1$ .

$1728 x^{3} - 432 x^{2} y z + 36 x y^{2} z^{2} + 1 x^{0} \cdot {(- y z)}^{3}$

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

$1728 x^{3} - 432 x^{2} y z + 36 x y^{2} z^{2} + x^{0} \cdot {(- y z)}^{3}$

Anything raised to $0$ is $1$ .

$1728 x^{3} - 432 x^{2} y z + 36 x y^{2} z^{2} + 1 \cdot {(- y z)}^{3}$

Multiply ${(- y z)}^{3}$ by $1$ .

$1728 x^{3} - 432 x^{2} y z + 36 x y^{2} z^{2} + {(- y z)}^{3}$

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $- y z$ .

$1728 x^{3} - 432 x^{2} y z + 36 x y^{2} z^{2} + {(- y)}^{3} z^{3}$

Apply the product rule to $- y$ .

$1728 x^{3} - 432 x^{2} y z + 36 x y^{2} z^{2} + {(- 1)}^{3} y^{3} z^{3}$

Raise $- 1$ to the power of $3$ .

$1728 x^{3} - 432 x^{2} y z + 36 x y^{2} z^{2} - y^{3} z^{3}$

Expand Using the Binomial Theorem (12x-yz)^3

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