Expand the Trigonometric Expression (4x-2)^4

${(4 x - 2)}^{4}$

Step 1

Use the Binomial Theorem.

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

Step 2

Simplify each term.

Apply the product rule to $4 x$ .

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

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

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

Apply the product rule to $4 x$ .

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

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

Move $4^{3}$ .

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

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

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

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

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

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

Add $3$ and $1$ .

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

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

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

Multiply $- 2$ by $256$ .

$256 x^{4} - 512 x^{3} + 6 {(4 x)}^{2} {(- 2)}^{2} + 4 (4 x) {(- 2)}^{3} + {(- 2)}^{4}$

Apply the product rule to $4 x$ .

$256 x^{4} - 512 x^{3} + 6 (4^{2} x^{2}) {(- 2)}^{2} + 4 (4 x) {(- 2)}^{3} + {(- 2)}^{4}$

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

$256 x^{4} - 512 x^{3} + 6 (16 x^{2}) {(- 2)}^{2} + 4 (4 x) {(- 2)}^{3} + {(- 2)}^{4}$

Multiply $16$ by $6$ .

$256 x^{4} - 512 x^{3} + 96 x^{2} {(- 2)}^{2} + 4 (4 x) {(- 2)}^{3} + {(- 2)}^{4}$

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

$256 x^{4} - 512 x^{3} + 96 x^{2} \cdot 4 + 4 (4 x) {(- 2)}^{3} + {(- 2)}^{4}$

Multiply $4$ by $96$ .

$256 x^{4} - 512 x^{3} + 384 x^{2} + 4 (4 x) {(- 2)}^{3} + {(- 2)}^{4}$

Multiply $4$ by $4$ .

$256 x^{4} - 512 x^{3} + 384 x^{2} + 16 x {(- 2)}^{3} + {(- 2)}^{4}$

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

$256 x^{4} - 512 x^{3} + 384 x^{2} + 16 x \cdot - 8 + {(- 2)}^{4}$

Multiply $- 8$ by $16$ .

$256 x^{4} - 512 x^{3} + 384 x^{2} - 128 x + {(- 2)}^{4}$

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

$256 x^{4} - 512 x^{3} + 384 x^{2} - 128 x + 16$