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

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

Step 1

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

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

Step 2

Expand the summation.

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

Step 3

Simplify the exponents for each term of the expansion.

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

Step 4

Simplify each term.

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

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

Apply the product rule to $4 x^{4}$ .

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

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

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

Multiply the exponents in ${(x^{4})}^{4}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Multiply $4$ by $4$ .

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

Anything raised to $0$ is $1$ .

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

Multiply $256$ by $1$ .

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

Apply the product rule to $4 x^{4}$ .

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

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

Move $4^{3}$ .

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

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

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

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

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

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

Add $3$ and $1$ .

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

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

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

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

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

Multiply the exponents in ${(x^{4})}^{3}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Multiply $4$ by $3$ .

$256 x^{16} + 256 \cdot x^{12} \cdot - 4 + 6 \cdot {(4 x^{4})}^{2} \cdot {(- 4)}^{2} + 4 \cdot {(4 x^{4})}^{1} \cdot {(- 4)}^{3} + 1 \cdot {(4 x^{4})}^{0} \cdot {(- 4)}^{4}$

Multiply $- 4$ by $256$ .

$256 x^{16} - 1024 x^{12} + 6 \cdot {(4 x^{4})}^{2} \cdot {(- 4)}^{2} + 4 \cdot {(4 x^{4})}^{1} \cdot {(- 4)}^{3} + 1 \cdot {(4 x^{4})}^{0} \cdot {(- 4)}^{4}$

Apply the product rule to $4 x^{4}$ .

$256 x^{16} - 1024 x^{12} + 6 \cdot (4^{2} {(x^{4})}^{2}) \cdot {(- 4)}^{2} + 4 \cdot {(4 x^{4})}^{1} \cdot {(- 4)}^{3} + 1 \cdot {(4 x^{4})}^{0} \cdot {(- 4)}^{4}$

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

$256 x^{16} - 1024 x^{12} + 6 \cdot (16 {(x^{4})}^{2}) \cdot {(- 4)}^{2} + 4 \cdot {(4 x^{4})}^{1} \cdot {(- 4)}^{3} + 1 \cdot {(4 x^{4})}^{0} \cdot {(- 4)}^{4}$

Multiply the exponents in ${(x^{4})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$256 x^{16} - 1024 x^{12} + 6 \cdot (16 x^{4 \cdot 2}) \cdot {(- 4)}^{2} + 4 \cdot {(4 x^{4})}^{1} \cdot {(- 4)}^{3} + 1 \cdot {(4 x^{4})}^{0} \cdot {(- 4)}^{4}$

Multiply $4$ by $2$ .

$256 x^{16} - 1024 x^{12} + 6 \cdot (16 x^{8}) \cdot {(- 4)}^{2} + 4 \cdot {(4 x^{4})}^{1} \cdot {(- 4)}^{3} + 1 \cdot {(4 x^{4})}^{0} \cdot {(- 4)}^{4}$

Multiply $16$ by $6$ .

$256 x^{16} - 1024 x^{12} + 96 x^{8} \cdot {(- 4)}^{2} + 4 \cdot {(4 x^{4})}^{1} \cdot {(- 4)}^{3} + 1 \cdot {(4 x^{4})}^{0} \cdot {(- 4)}^{4}$

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

$256 x^{16} - 1024 x^{12} + 96 x^{8} \cdot 16 + 4 \cdot {(4 x^{4})}^{1} \cdot {(- 4)}^{3} + 1 \cdot {(4 x^{4})}^{0} \cdot {(- 4)}^{4}$

Multiply $16$ by $96$ .

$256 x^{16} - 1024 x^{12} + 1536 x^{8} + 4 \cdot {(4 x^{4})}^{1} \cdot {(- 4)}^{3} + 1 \cdot {(4 x^{4})}^{0} \cdot {(- 4)}^{4}$

Simplify.

$256 x^{16} - 1024 x^{12} + 1536 x^{8} + 4 \cdot (4 x^{4}) \cdot {(- 4)}^{3} + 1 \cdot {(4 x^{4})}^{0} \cdot {(- 4)}^{4}$

Multiply $4$ by $4$ .

$256 x^{16} - 1024 x^{12} + 1536 x^{8} + 16 x^{4} \cdot {(- 4)}^{3} + 1 \cdot {(4 x^{4})}^{0} \cdot {(- 4)}^{4}$

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

$256 x^{16} - 1024 x^{12} + 1536 x^{8} + 16 x^{4} \cdot - 64 + 1 \cdot {(4 x^{4})}^{0} \cdot {(- 4)}^{4}$

Multiply $- 64$ by $16$ .

$256 x^{16} - 1024 x^{12} + 1536 x^{8} - 1024 x^{4} + 1 \cdot {(4 x^{4})}^{0} \cdot {(- 4)}^{4}$

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

$256 x^{16} - 1024 x^{12} + 1536 x^{8} - 1024 x^{4} + {(4 x^{4})}^{0} \cdot {(- 4)}^{4}$

Apply the product rule to $4 x^{4}$ .

$256 x^{16} - 1024 x^{12} + 1536 x^{8} - 1024 x^{4} + 4^{0} {(x^{4})}^{0} \cdot {(- 4)}^{4}$

Anything raised to $0$ is $1$ .

$256 x^{16} - 1024 x^{12} + 1536 x^{8} - 1024 x^{4} + 1 {(x^{4})}^{0} \cdot {(- 4)}^{4}$

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

$256 x^{16} - 1024 x^{12} + 1536 x^{8} - 1024 x^{4} + {(x^{4})}^{0} \cdot {(- 4)}^{4}$

Multiply the exponents in ${(x^{4})}^{0}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$256 x^{16} - 1024 x^{12} + 1536 x^{8} - 1024 x^{4} + x^{4 \cdot 0} \cdot {(- 4)}^{4}$

Multiply $4$ by $0$ .

$256 x^{16} - 1024 x^{12} + 1536 x^{8} - 1024 x^{4} + x^{0} \cdot {(- 4)}^{4}$

Anything raised to $0$ is $1$ .

$256 x^{16} - 1024 x^{12} + 1536 x^{8} - 1024 x^{4} + 1 \cdot {(- 4)}^{4}$

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

$256 x^{16} - 1024 x^{12} + 1536 x^{8} - 1024 x^{4} + {(- 4)}^{4}$

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

$256 x^{16} - 1024 x^{12} + 1536 x^{8} - 1024 x^{4} + 256$

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

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