Expand Using the Binomial Theorem (4x-4y)^4

${(4 x - 4 y)}^{4}$

Step 1

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

$\sum_{k = 0}^{4} \frac{4!}{(4 - k)! k!} \cdot {(4 x)}^{4 - k} \cdot {(- 4 y)}^{k}$

Step 2

Expand the summation.

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

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(4 x)}^{4} \cdot {(- 4 y)}^{0} + 4 \cdot {(4 x)}^{3} \cdot {(- 4 y)}^{1} + 6 \cdot {(4 x)}^{2} \cdot {(- 4 y)}^{2} + 4 \cdot {(4 x)}^{1} \cdot {(- 4 y)}^{3} + 1 \cdot {(4 x)}^{0} \cdot {(- 4 y)}^{4}$

Step 4

Simplify each term.

Multiply ${(4 x)}^{4}$ by $1$ .

${(4 x)}^{4} \cdot {(- 4 y)}^{0} + 4 \cdot {(4 x)}^{3} \cdot {(- 4 y)}^{1} + 6 \cdot {(4 x)}^{2} \cdot {(- 4 y)}^{2} + 4 \cdot {(4 x)}^{1} \cdot {(- 4 y)}^{3} + 1 \cdot {(4 x)}^{0} \cdot {(- 4 y)}^{4}$

Apply the product rule to $4 x$ .

$4^{4} x^{4} \cdot {(- 4 y)}^{0} + 4 \cdot {(4 x)}^{3} \cdot {(- 4 y)}^{1} + 6 \cdot {(4 x)}^{2} \cdot {(- 4 y)}^{2} + 4 \cdot {(4 x)}^{1} \cdot {(- 4 y)}^{3} + 1 \cdot {(4 x)}^{0} \cdot {(- 4 y)}^{4}$

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

$256 x^{4} \cdot {(- 4 y)}^{0} + 4 \cdot {(4 x)}^{3} \cdot {(- 4 y)}^{1} + 6 \cdot {(4 x)}^{2} \cdot {(- 4 y)}^{2} + 4 \cdot {(4 x)}^{1} \cdot {(- 4 y)}^{3} + 1 \cdot {(4 x)}^{0} \cdot {(- 4 y)}^{4}$

Apply the product rule to $- 4 y$ .

$256 x^{4} \cdot ({(- 4)}^{0} y^{0}) + 4 \cdot {(4 x)}^{3} \cdot {(- 4 y)}^{1} + 6 \cdot {(4 x)}^{2} \cdot {(- 4 y)}^{2} + 4 \cdot {(4 x)}^{1} \cdot {(- 4 y)}^{3} + 1 \cdot {(4 x)}^{0} \cdot {(- 4 y)}^{4}$

Rewrite using the commutative property of multiplication.

$256 \cdot {(- 4)}^{0} x^{4} y^{0} + 4 \cdot {(4 x)}^{3} \cdot {(- 4 y)}^{1} + 6 \cdot {(4 x)}^{2} \cdot {(- 4 y)}^{2} + 4 \cdot {(4 x)}^{1} \cdot {(- 4 y)}^{3} + 1 \cdot {(4 x)}^{0} \cdot {(- 4 y)}^{4}$

Anything raised to $0$ is $1$ .

$256 \cdot 1 x^{4} y^{0} + 4 \cdot {(4 x)}^{3} \cdot {(- 4 y)}^{1} + 6 \cdot {(4 x)}^{2} \cdot {(- 4 y)}^{2} + 4 \cdot {(4 x)}^{1} \cdot {(- 4 y)}^{3} + 1 \cdot {(4 x)}^{0} \cdot {(- 4 y)}^{4}$

Multiply $256$ by $1$ .

$256 x^{4} y^{0} + 4 \cdot {(4 x)}^{3} \cdot {(- 4 y)}^{1} + 6 \cdot {(4 x)}^{2} \cdot {(- 4 y)}^{2} + 4 \cdot {(4 x)}^{1} \cdot {(- 4 y)}^{3} + 1 \cdot {(4 x)}^{0} \cdot {(- 4 y)}^{4}$

Anything raised to $0$ is $1$ .

$256 x^{4} \cdot 1 + 4 \cdot {(4 x)}^{3} \cdot {(- 4 y)}^{1} + 6 \cdot {(4 x)}^{2} \cdot {(- 4 y)}^{2} + 4 \cdot {(4 x)}^{1} \cdot {(- 4 y)}^{3} + 1 \cdot {(4 x)}^{0} \cdot {(- 4 y)}^{4}$

Multiply $256$ by $1$ .

$256 x^{4} + 4 \cdot {(4 x)}^{3} \cdot {(- 4 y)}^{1} + 6 \cdot {(4 x)}^{2} \cdot {(- 4 y)}^{2} + 4 \cdot {(4 x)}^{1} \cdot {(- 4 y)}^{3} + 1 \cdot {(4 x)}^{0} \cdot {(- 4 y)}^{4}$

Apply the product rule to $4 x$ .

$256 x^{4} + 4 \cdot (4^{3} x^{3}) \cdot {(- 4 y)}^{1} + 6 \cdot {(4 x)}^{2} \cdot {(- 4 y)}^{2} + 4 \cdot {(4 x)}^{1} \cdot {(- 4 y)}^{3} + 1 \cdot {(4 x)}^{0} \cdot {(- 4 y)}^{4}$

Multiply $4$ by $4^{3}$ by adding the exponents.

Move $4^{3}$ .

$256 x^{4} + 4^{3} \cdot 4 \cdot x^{3} \cdot {(- 4 y)}^{1} + 6 \cdot {(4 x)}^{2} \cdot {(- 4 y)}^{2} + 4 \cdot {(4 x)}^{1} \cdot {(- 4 y)}^{3} + 1 \cdot {(4 x)}^{0} \cdot {(- 4 y)}^{4}$

Multiply $4^{3}$ by $4$ .

Raise $4$ to the power of $1$ .

$256 x^{4} + 4^{3} \cdot 4^{1} \cdot x^{3} \cdot {(- 4 y)}^{1} + 6 \cdot {(4 x)}^{2} \cdot {(- 4 y)}^{2} + 4 \cdot {(4 x)}^{1} \cdot {(- 4 y)}^{3} + 1 \cdot {(4 x)}^{0} \cdot {(- 4 y)}^{4}$

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

$256 x^{4} + 4^{3 + 1} \cdot x^{3} \cdot {(- 4 y)}^{1} + 6 \cdot {(4 x)}^{2} \cdot {(- 4 y)}^{2} + 4 \cdot {(4 x)}^{1} \cdot {(- 4 y)}^{3} + 1 \cdot {(4 x)}^{0} \cdot {(- 4 y)}^{4}$

Add $3$ and $1$ .

$256 x^{4} + 4^{4} \cdot x^{3} \cdot {(- 4 y)}^{1} + 6 \cdot {(4 x)}^{2} \cdot {(- 4 y)}^{2} + 4 \cdot {(4 x)}^{1} \cdot {(- 4 y)}^{3} + 1 \cdot {(4 x)}^{0} \cdot {(- 4 y)}^{4}$

Simplify $4^{4} \cdot x^{3} \cdot {(- 4 y)}^{1}$ .

$256 x^{4} + 4^{4} \cdot x^{3} \cdot (- 4 y) + 6 \cdot {(4 x)}^{2} \cdot {(- 4 y)}^{2} + 4 \cdot {(4 x)}^{1} \cdot {(- 4 y)}^{3} + 1 \cdot {(4 x)}^{0} \cdot {(- 4 y)}^{4}$

Rewrite using the commutative property of multiplication.

$256 x^{4} + 4^{4} \cdot - 4 x^{3} y + 6 \cdot {(4 x)}^{2} \cdot {(- 4 y)}^{2} + 4 \cdot {(4 x)}^{1} \cdot {(- 4 y)}^{3} + 1 \cdot {(4 x)}^{0} \cdot {(- 4 y)}^{4}$

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

$256 x^{4} + 256 \cdot - 4 x^{3} y + 6 \cdot {(4 x)}^{2} \cdot {(- 4 y)}^{2} + 4 \cdot {(4 x)}^{1} \cdot {(- 4 y)}^{3} + 1 \cdot {(4 x)}^{0} \cdot {(- 4 y)}^{4}$

Multiply $256$ by $- 4$ .

$256 x^{4} - 1024 x^{3} y + 6 \cdot {(4 x)}^{2} \cdot {(- 4 y)}^{2} + 4 \cdot {(4 x)}^{1} \cdot {(- 4 y)}^{3} + 1 \cdot {(4 x)}^{0} \cdot {(- 4 y)}^{4}$

Apply the product rule to $4 x$ .

$256 x^{4} - 1024 x^{3} y + 6 \cdot (4^{2} x^{2}) \cdot {(- 4 y)}^{2} + 4 \cdot {(4 x)}^{1} \cdot {(- 4 y)}^{3} + 1 \cdot {(4 x)}^{0} \cdot {(- 4 y)}^{4}$

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

$256 x^{4} - 1024 x^{3} y + 6 \cdot (16 x^{2}) \cdot {(- 4 y)}^{2} + 4 \cdot {(4 x)}^{1} \cdot {(- 4 y)}^{3} + 1 \cdot {(4 x)}^{0} \cdot {(- 4 y)}^{4}$

Multiply $16$ by $6$ .

$256 x^{4} - 1024 x^{3} y + 96 x^{2} \cdot {(- 4 y)}^{2} + 4 \cdot {(4 x)}^{1} \cdot {(- 4 y)}^{3} + 1 \cdot {(4 x)}^{0} \cdot {(- 4 y)}^{4}$

Apply the product rule to $- 4 y$ .

$256 x^{4} - 1024 x^{3} y + 96 x^{2} \cdot ({(- 4)}^{2} y^{2}) + 4 \cdot {(4 x)}^{1} \cdot {(- 4 y)}^{3} + 1 \cdot {(4 x)}^{0} \cdot {(- 4 y)}^{4}$

Rewrite using the commutative property of multiplication.

$256 x^{4} - 1024 x^{3} y + 96 \cdot {(- 4)}^{2} x^{2} y^{2} + 4 \cdot {(4 x)}^{1} \cdot {(- 4 y)}^{3} + 1 \cdot {(4 x)}^{0} \cdot {(- 4 y)}^{4}$

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

$256 x^{4} - 1024 x^{3} y + 96 \cdot 16 x^{2} y^{2} + 4 \cdot {(4 x)}^{1} \cdot {(- 4 y)}^{3} + 1 \cdot {(4 x)}^{0} \cdot {(- 4 y)}^{4}$

Multiply $96$ by $16$ .

$256 x^{4} - 1024 x^{3} y + 1536 x^{2} y^{2} + 4 \cdot {(4 x)}^{1} \cdot {(- 4 y)}^{3} + 1 \cdot {(4 x)}^{0} \cdot {(- 4 y)}^{4}$

Simplify.

$256 x^{4} - 1024 x^{3} y + 1536 x^{2} y^{2} + 4 \cdot (4 x) \cdot {(- 4 y)}^{3} + 1 \cdot {(4 x)}^{0} \cdot {(- 4 y)}^{4}$

Multiply $4$ by $4$ .

$256 x^{4} - 1024 x^{3} y + 1536 x^{2} y^{2} + 16 x \cdot {(- 4 y)}^{3} + 1 \cdot {(4 x)}^{0} \cdot {(- 4 y)}^{4}$

Apply the product rule to $- 4 y$ .

$256 x^{4} - 1024 x^{3} y + 1536 x^{2} y^{2} + 16 x \cdot ({(- 4)}^{3} y^{3}) + 1 \cdot {(4 x)}^{0} \cdot {(- 4 y)}^{4}$

Rewrite using the commutative property of multiplication.

$256 x^{4} - 1024 x^{3} y + 1536 x^{2} y^{2} + 16 \cdot {(- 4)}^{3} x y^{3} + 1 \cdot {(4 x)}^{0} \cdot {(- 4 y)}^{4}$

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

$256 x^{4} - 1024 x^{3} y + 1536 x^{2} y^{2} + 16 \cdot - 64 x y^{3} + 1 \cdot {(4 x)}^{0} \cdot {(- 4 y)}^{4}$

Multiply $16$ by $- 64$ .

$256 x^{4} - 1024 x^{3} y + 1536 x^{2} y^{2} - 1024 x y^{3} + 1 \cdot {(4 x)}^{0} \cdot {(- 4 y)}^{4}$

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

$256 x^{4} - 1024 x^{3} y + 1536 x^{2} y^{2} - 1024 x y^{3} + {(4 x)}^{0} \cdot {(- 4 y)}^{4}$

Apply the product rule to $4 x$ .

$256 x^{4} - 1024 x^{3} y + 1536 x^{2} y^{2} - 1024 x y^{3} + 4^{0} x^{0} \cdot {(- 4 y)}^{4}$

Anything raised to $0$ is $1$ .

$256 x^{4} - 1024 x^{3} y + 1536 x^{2} y^{2} - 1024 x y^{3} + 1 x^{0} \cdot {(- 4 y)}^{4}$

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

$256 x^{4} - 1024 x^{3} y + 1536 x^{2} y^{2} - 1024 x y^{3} + x^{0} \cdot {(- 4 y)}^{4}$

Anything raised to $0$ is $1$ .

$256 x^{4} - 1024 x^{3} y + 1536 x^{2} y^{2} - 1024 x y^{3} + 1 \cdot {(- 4 y)}^{4}$

Multiply ${(- 4 y)}^{4}$ by $1$ .

$256 x^{4} - 1024 x^{3} y + 1536 x^{2} y^{2} - 1024 x y^{3} + {(- 4 y)}^{4}$

Apply the product rule to $- 4 y$ .

$256 x^{4} - 1024 x^{3} y + 1536 x^{2} y^{2} - 1024 x y^{3} + {(- 4)}^{4} y^{4}$

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

$256 x^{4} - 1024 x^{3} y + 1536 x^{2} y^{2} - 1024 x y^{3} + 256 y^{4}$

Expand Using the Binomial Theorem (4x-4y)^4

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