Expand the Trigonometric Expression (4x-3)^8

${(4 x - 3)}^{8}$

Step 1

Use the Binomial Theorem.

${(4 x)}^{8} + 8 {(4 x)}^{7} \cdot - 3 + 28 {(4 x)}^{6} {(- 3)}^{2} + 56 {(4 x)}^{5} {(- 3)}^{3} + 70 {(4 x)}^{4} {(- 3)}^{4} + 56 {(4 x)}^{3} {(- 3)}^{5} + 28 {(4 x)}^{2} {(- 3)}^{6} + 8 (4 x) {(- 3)}^{7} + {(- 3)}^{8}$

Step 2

Simplify each term.

Apply the product rule to $4 x$ .

$4^{8} x^{8} + 8 {(4 x)}^{7} \cdot - 3 + 28 {(4 x)}^{6} {(- 3)}^{2} + 56 {(4 x)}^{5} {(- 3)}^{3} + 70 {(4 x)}^{4} {(- 3)}^{4} + 56 {(4 x)}^{3} {(- 3)}^{5} + 28 {(4 x)}^{2} {(- 3)}^{6} + 8 (4 x) {(- 3)}^{7} + {(- 3)}^{8}$

Raise $4$ to the power of $8$ .

$65536 x^{8} + 8 {(4 x)}^{7} \cdot - 3 + 28 {(4 x)}^{6} {(- 3)}^{2} + 56 {(4 x)}^{5} {(- 3)}^{3} + 70 {(4 x)}^{4} {(- 3)}^{4} + 56 {(4 x)}^{3} {(- 3)}^{5} + 28 {(4 x)}^{2} {(- 3)}^{6} + 8 (4 x) {(- 3)}^{7} + {(- 3)}^{8}$

Apply the product rule to $4 x$ .

$65536 x^{8} + 8 (4^{7} x^{7}) \cdot - 3 + 28 {(4 x)}^{6} {(- 3)}^{2} + 56 {(4 x)}^{5} {(- 3)}^{3} + 70 {(4 x)}^{4} {(- 3)}^{4} + 56 {(4 x)}^{3} {(- 3)}^{5} + 28 {(4 x)}^{2} {(- 3)}^{6} + 8 (4 x) {(- 3)}^{7} + {(- 3)}^{8}$

Raise $4$ to the power of $7$ .

$65536 x^{8} + 8 (16384 x^{7}) \cdot - 3 + 28 {(4 x)}^{6} {(- 3)}^{2} + 56 {(4 x)}^{5} {(- 3)}^{3} + 70 {(4 x)}^{4} {(- 3)}^{4} + 56 {(4 x)}^{3} {(- 3)}^{5} + 28 {(4 x)}^{2} {(- 3)}^{6} + 8 (4 x) {(- 3)}^{7} + {(- 3)}^{8}$

Multiply $16384$ by $8$ .

$65536 x^{8} + 131072 x^{7} \cdot - 3 + 28 {(4 x)}^{6} {(- 3)}^{2} + 56 {(4 x)}^{5} {(- 3)}^{3} + 70 {(4 x)}^{4} {(- 3)}^{4} + 56 {(4 x)}^{3} {(- 3)}^{5} + 28 {(4 x)}^{2} {(- 3)}^{6} + 8 (4 x) {(- 3)}^{7} + {(- 3)}^{8}$

Multiply $- 3$ by $131072$ .

$65536 x^{8} - 393216 x^{7} + 28 {(4 x)}^{6} {(- 3)}^{2} + 56 {(4 x)}^{5} {(- 3)}^{3} + 70 {(4 x)}^{4} {(- 3)}^{4} + 56 {(4 x)}^{3} {(- 3)}^{5} + 28 {(4 x)}^{2} {(- 3)}^{6} + 8 (4 x) {(- 3)}^{7} + {(- 3)}^{8}$

Apply the product rule to $4 x$ .

$65536 x^{8} - 393216 x^{7} + 28 (4^{6} x^{6}) {(- 3)}^{2} + 56 {(4 x)}^{5} {(- 3)}^{3} + 70 {(4 x)}^{4} {(- 3)}^{4} + 56 {(4 x)}^{3} {(- 3)}^{5} + 28 {(4 x)}^{2} {(- 3)}^{6} + 8 (4 x) {(- 3)}^{7} + {(- 3)}^{8}$

Raise $4$ to the power of $6$ .

$65536 x^{8} - 393216 x^{7} + 28 (4096 x^{6}) {(- 3)}^{2} + 56 {(4 x)}^{5} {(- 3)}^{3} + 70 {(4 x)}^{4} {(- 3)}^{4} + 56 {(4 x)}^{3} {(- 3)}^{5} + 28 {(4 x)}^{2} {(- 3)}^{6} + 8 (4 x) {(- 3)}^{7} + {(- 3)}^{8}$

Multiply $4096$ by $28$ .

$65536 x^{8} - 393216 x^{7} + 114688 x^{6} {(- 3)}^{2} + 56 {(4 x)}^{5} {(- 3)}^{3} + 70 {(4 x)}^{4} {(- 3)}^{4} + 56 {(4 x)}^{3} {(- 3)}^{5} + 28 {(4 x)}^{2} {(- 3)}^{6} + 8 (4 x) {(- 3)}^{7} + {(- 3)}^{8}$

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

$65536 x^{8} - 393216 x^{7} + 114688 x^{6} \cdot 9 + 56 {(4 x)}^{5} {(- 3)}^{3} + 70 {(4 x)}^{4} {(- 3)}^{4} + 56 {(4 x)}^{3} {(- 3)}^{5} + 28 {(4 x)}^{2} {(- 3)}^{6} + 8 (4 x) {(- 3)}^{7} + {(- 3)}^{8}$

Multiply $9$ by $114688$ .

$65536 x^{8} - 393216 x^{7} + 1032192 x^{6} + 56 {(4 x)}^{5} {(- 3)}^{3} + 70 {(4 x)}^{4} {(- 3)}^{4} + 56 {(4 x)}^{3} {(- 3)}^{5} + 28 {(4 x)}^{2} {(- 3)}^{6} + 8 (4 x) {(- 3)}^{7} + {(- 3)}^{8}$

Apply the product rule to $4 x$ .

$65536 x^{8} - 393216 x^{7} + 1032192 x^{6} + 56 (4^{5} x^{5}) {(- 3)}^{3} + 70 {(4 x)}^{4} {(- 3)}^{4} + 56 {(4 x)}^{3} {(- 3)}^{5} + 28 {(4 x)}^{2} {(- 3)}^{6} + 8 (4 x) {(- 3)}^{7} + {(- 3)}^{8}$

Raise $4$ to the power of $5$ .

$65536 x^{8} - 393216 x^{7} + 1032192 x^{6} + 56 (1024 x^{5}) {(- 3)}^{3} + 70 {(4 x)}^{4} {(- 3)}^{4} + 56 {(4 x)}^{3} {(- 3)}^{5} + 28 {(4 x)}^{2} {(- 3)}^{6} + 8 (4 x) {(- 3)}^{7} + {(- 3)}^{8}$

Multiply $1024$ by $56$ .

$65536 x^{8} - 393216 x^{7} + 1032192 x^{6} + 57344 x^{5} {(- 3)}^{3} + 70 {(4 x)}^{4} {(- 3)}^{4} + 56 {(4 x)}^{3} {(- 3)}^{5} + 28 {(4 x)}^{2} {(- 3)}^{6} + 8 (4 x) {(- 3)}^{7} + {(- 3)}^{8}$

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

$65536 x^{8} - 393216 x^{7} + 1032192 x^{6} + 57344 x^{5} \cdot - 27 + 70 {(4 x)}^{4} {(- 3)}^{4} + 56 {(4 x)}^{3} {(- 3)}^{5} + 28 {(4 x)}^{2} {(- 3)}^{6} + 8 (4 x) {(- 3)}^{7} + {(- 3)}^{8}$

Multiply $- 27$ by $57344$ .

$65536 x^{8} - 393216 x^{7} + 1032192 x^{6} - 1548288 x^{5} + 70 {(4 x)}^{4} {(- 3)}^{4} + 56 {(4 x)}^{3} {(- 3)}^{5} + 28 {(4 x)}^{2} {(- 3)}^{6} + 8 (4 x) {(- 3)}^{7} + {(- 3)}^{8}$

Apply the product rule to $4 x$ .

$65536 x^{8} - 393216 x^{7} + 1032192 x^{6} - 1548288 x^{5} + 70 (4^{4} x^{4}) {(- 3)}^{4} + 56 {(4 x)}^{3} {(- 3)}^{5} + 28 {(4 x)}^{2} {(- 3)}^{6} + 8 (4 x) {(- 3)}^{7} + {(- 3)}^{8}$

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

$65536 x^{8} - 393216 x^{7} + 1032192 x^{6} - 1548288 x^{5} + 70 (256 x^{4}) {(- 3)}^{4} + 56 {(4 x)}^{3} {(- 3)}^{5} + 28 {(4 x)}^{2} {(- 3)}^{6} + 8 (4 x) {(- 3)}^{7} + {(- 3)}^{8}$

Multiply $256$ by $70$ .

$65536 x^{8} - 393216 x^{7} + 1032192 x^{6} - 1548288 x^{5} + 17920 x^{4} {(- 3)}^{4} + 56 {(4 x)}^{3} {(- 3)}^{5} + 28 {(4 x)}^{2} {(- 3)}^{6} + 8 (4 x) {(- 3)}^{7} + {(- 3)}^{8}$

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

$65536 x^{8} - 393216 x^{7} + 1032192 x^{6} - 1548288 x^{5} + 17920 x^{4} \cdot 81 + 56 {(4 x)}^{3} {(- 3)}^{5} + 28 {(4 x)}^{2} {(- 3)}^{6} + 8 (4 x) {(- 3)}^{7} + {(- 3)}^{8}$

Multiply $81$ by $17920$ .

$65536 x^{8} - 393216 x^{7} + 1032192 x^{6} - 1548288 x^{5} + 1451520 x^{4} + 56 {(4 x)}^{3} {(- 3)}^{5} + 28 {(4 x)}^{2} {(- 3)}^{6} + 8 (4 x) {(- 3)}^{7} + {(- 3)}^{8}$

Apply the product rule to $4 x$ .

$65536 x^{8} - 393216 x^{7} + 1032192 x^{6} - 1548288 x^{5} + 1451520 x^{4} + 56 (4^{3} x^{3}) {(- 3)}^{5} + 28 {(4 x)}^{2} {(- 3)}^{6} + 8 (4 x) {(- 3)}^{7} + {(- 3)}^{8}$

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

$65536 x^{8} - 393216 x^{7} + 1032192 x^{6} - 1548288 x^{5} + 1451520 x^{4} + 56 (64 x^{3}) {(- 3)}^{5} + 28 {(4 x)}^{2} {(- 3)}^{6} + 8 (4 x) {(- 3)}^{7} + {(- 3)}^{8}$

Multiply $64$ by $56$ .

$65536 x^{8} - 393216 x^{7} + 1032192 x^{6} - 1548288 x^{5} + 1451520 x^{4} + 3584 x^{3} {(- 3)}^{5} + 28 {(4 x)}^{2} {(- 3)}^{6} + 8 (4 x) {(- 3)}^{7} + {(- 3)}^{8}$

Raise $- 3$ to the power of $5$ .

$65536 x^{8} - 393216 x^{7} + 1032192 x^{6} - 1548288 x^{5} + 1451520 x^{4} + 3584 x^{3} \cdot - 243 + 28 {(4 x)}^{2} {(- 3)}^{6} + 8 (4 x) {(- 3)}^{7} + {(- 3)}^{8}$

Multiply $- 243$ by $3584$ .

$65536 x^{8} - 393216 x^{7} + 1032192 x^{6} - 1548288 x^{5} + 1451520 x^{4} - 870912 x^{3} + 28 {(4 x)}^{2} {(- 3)}^{6} + 8 (4 x) {(- 3)}^{7} + {(- 3)}^{8}$

Apply the product rule to $4 x$ .

$65536 x^{8} - 393216 x^{7} + 1032192 x^{6} - 1548288 x^{5} + 1451520 x^{4} - 870912 x^{3} + 28 (4^{2} x^{2}) {(- 3)}^{6} + 8 (4 x) {(- 3)}^{7} + {(- 3)}^{8}$

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

$65536 x^{8} - 393216 x^{7} + 1032192 x^{6} - 1548288 x^{5} + 1451520 x^{4} - 870912 x^{3} + 28 (16 x^{2}) {(- 3)}^{6} + 8 (4 x) {(- 3)}^{7} + {(- 3)}^{8}$

Multiply $16$ by $28$ .

$65536 x^{8} - 393216 x^{7} + 1032192 x^{6} - 1548288 x^{5} + 1451520 x^{4} - 870912 x^{3} + 448 x^{2} {(- 3)}^{6} + 8 (4 x) {(- 3)}^{7} + {(- 3)}^{8}$

Raise $- 3$ to the power of $6$ .

$65536 x^{8} - 393216 x^{7} + 1032192 x^{6} - 1548288 x^{5} + 1451520 x^{4} - 870912 x^{3} + 448 x^{2} \cdot 729 + 8 (4 x) {(- 3)}^{7} + {(- 3)}^{8}$

Multiply $729$ by $448$ .

$65536 x^{8} - 393216 x^{7} + 1032192 x^{6} - 1548288 x^{5} + 1451520 x^{4} - 870912 x^{3} + 326592 x^{2} + 8 (4 x) {(- 3)}^{7} + {(- 3)}^{8}$

Multiply $4$ by $8$ .

$65536 x^{8} - 393216 x^{7} + 1032192 x^{6} - 1548288 x^{5} + 1451520 x^{4} - 870912 x^{3} + 326592 x^{2} + 32 x {(- 3)}^{7} + {(- 3)}^{8}$

Raise $- 3$ to the power of $7$ .

$65536 x^{8} - 393216 x^{7} + 1032192 x^{6} - 1548288 x^{5} + 1451520 x^{4} - 870912 x^{3} + 326592 x^{2} + 32 x \cdot - 2187 + {(- 3)}^{8}$

Multiply $- 2187$ by $32$ .

$65536 x^{8} - 393216 x^{7} + 1032192 x^{6} - 1548288 x^{5} + 1451520 x^{4} - 870912 x^{3} + 326592 x^{2} - 69984 x + {(- 3)}^{8}$

Raise $- 3$ to the power of $8$ .

$65536 x^{8} - 393216 x^{7} + 1032192 x^{6} - 1548288 x^{5} + 1451520 x^{4} - 870912 x^{3} + 326592 x^{2} - 69984 x + 6561$

Expand the Trigonometric Expression (4x-3)^8

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