Expand the Trigonometric Expression (4y-1)^4

${(4 y - 1)}^{4}$

Step 1

Use the Binomial Theorem.

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

Step 2

Simplify each term.

Apply the product rule to $4 y$ .

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

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

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

Apply the product rule to $4 y$ .

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

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

Move $4^{3}$ .

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

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

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

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

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

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

Add $3$ and $1$ .

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

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

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

Multiply $- 1$ by $256$ .

$256 y^{4} - 256 y^{3} + 6 {(4 y)}^{2} {(- 1)}^{2} + 4 (4 y) {(- 1)}^{3} + {(- 1)}^{4}$

Apply the product rule to $4 y$ .

$256 y^{4} - 256 y^{3} + 6 (4^{2} y^{2}) {(- 1)}^{2} + 4 (4 y) {(- 1)}^{3} + {(- 1)}^{4}$

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

$256 y^{4} - 256 y^{3} + 6 (16 y^{2}) {(- 1)}^{2} + 4 (4 y) {(- 1)}^{3} + {(- 1)}^{4}$

Multiply $16$ by $6$ .

$256 y^{4} - 256 y^{3} + 96 y^{2} {(- 1)}^{2} + 4 (4 y) {(- 1)}^{3} + {(- 1)}^{4}$

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

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

Multiply $96$ by $1$ .

$256 y^{4} - 256 y^{3} + 96 y^{2} + 4 (4 y) {(- 1)}^{3} + {(- 1)}^{4}$

Multiply $4$ by $4$ .

$256 y^{4} - 256 y^{3} + 96 y^{2} + 16 y {(- 1)}^{3} + {(- 1)}^{4}$

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

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

Multiply $- 1$ by $16$ .

$256 y^{4} - 256 y^{3} + 96 y^{2} - 16 y + {(- 1)}^{4}$

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

$256 y^{4} - 256 y^{3} + 96 y^{2} - 16 y + 1$