Expand the Trigonometric Expression (4x+5y)^6

${(4 x + 5 y)}^{6}$

Step 1

Use the Binomial Theorem.

${(4 x)}^{6} + 6 {(4 x)}^{5} (5 y) + 15 {(4 x)}^{4} {(5 y)}^{2} + 20 {(4 x)}^{3} {(5 y)}^{3} + 15 {(4 x)}^{2} {(5 y)}^{4} + 6 (4 x) {(5 y)}^{5} + {(5 y)}^{6}$

Step 2

Simplify each term.

Apply the product rule to $4 x$ .

$4^{6} x^{6} + 6 {(4 x)}^{5} (5 y) + 15 {(4 x)}^{4} {(5 y)}^{2} + 20 {(4 x)}^{3} {(5 y)}^{3} + 15 {(4 x)}^{2} {(5 y)}^{4} + 6 (4 x) {(5 y)}^{5} + {(5 y)}^{6}$

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

$4096 x^{6} + 6 {(4 x)}^{5} (5 y) + 15 {(4 x)}^{4} {(5 y)}^{2} + 20 {(4 x)}^{3} {(5 y)}^{3} + 15 {(4 x)}^{2} {(5 y)}^{4} + 6 (4 x) {(5 y)}^{5} + {(5 y)}^{6}$

Rewrite using the commutative property of multiplication.

$4096 x^{6} + 6 \cdot 5 {(4 x)}^{5} y + 15 {(4 x)}^{4} {(5 y)}^{2} + 20 {(4 x)}^{3} {(5 y)}^{3} + 15 {(4 x)}^{2} {(5 y)}^{4} + 6 (4 x) {(5 y)}^{5} + {(5 y)}^{6}$

Multiply $6$ by $5$ .

$4096 x^{6} + 30 {(4 x)}^{5} y + 15 {(4 x)}^{4} {(5 y)}^{2} + 20 {(4 x)}^{3} {(5 y)}^{3} + 15 {(4 x)}^{2} {(5 y)}^{4} + 6 (4 x) {(5 y)}^{5} + {(5 y)}^{6}$

Apply the product rule to $4 x$ .

$4096 x^{6} + 30 (4^{5} x^{5}) y + 15 {(4 x)}^{4} {(5 y)}^{2} + 20 {(4 x)}^{3} {(5 y)}^{3} + 15 {(4 x)}^{2} {(5 y)}^{4} + 6 (4 x) {(5 y)}^{5} + {(5 y)}^{6}$

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

$4096 x^{6} + 30 (1024 x^{5}) y + 15 {(4 x)}^{4} {(5 y)}^{2} + 20 {(4 x)}^{3} {(5 y)}^{3} + 15 {(4 x)}^{2} {(5 y)}^{4} + 6 (4 x) {(5 y)}^{5} + {(5 y)}^{6}$

Multiply $1024$ by $30$ .

$4096 x^{6} + 30720 x^{5} y + 15 {(4 x)}^{4} {(5 y)}^{2} + 20 {(4 x)}^{3} {(5 y)}^{3} + 15 {(4 x)}^{2} {(5 y)}^{4} + 6 (4 x) {(5 y)}^{5} + {(5 y)}^{6}$

Apply the product rule to $4 x$ .

$4096 x^{6} + 30720 x^{5} y + 15 (4^{4} x^{4}) {(5 y)}^{2} + 20 {(4 x)}^{3} {(5 y)}^{3} + 15 {(4 x)}^{2} {(5 y)}^{4} + 6 (4 x) {(5 y)}^{5} + {(5 y)}^{6}$

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

$4096 x^{6} + 30720 x^{5} y + 15 (256 x^{4}) {(5 y)}^{2} + 20 {(4 x)}^{3} {(5 y)}^{3} + 15 {(4 x)}^{2} {(5 y)}^{4} + 6 (4 x) {(5 y)}^{5} + {(5 y)}^{6}$

Multiply $256$ by $15$ .

$4096 x^{6} + 30720 x^{5} y + 3840 x^{4} {(5 y)}^{2} + 20 {(4 x)}^{3} {(5 y)}^{3} + 15 {(4 x)}^{2} {(5 y)}^{4} + 6 (4 x) {(5 y)}^{5} + {(5 y)}^{6}$

Apply the product rule to $5 y$ .

$4096 x^{6} + 30720 x^{5} y + 3840 x^{4} (5^{2} y^{2}) + 20 {(4 x)}^{3} {(5 y)}^{3} + 15 {(4 x)}^{2} {(5 y)}^{4} + 6 (4 x) {(5 y)}^{5} + {(5 y)}^{6}$

Rewrite using the commutative property of multiplication.

$4096 x^{6} + 30720 x^{5} y + 3840 \cdot 5^{2} x^{4} y^{2} + 20 {(4 x)}^{3} {(5 y)}^{3} + 15 {(4 x)}^{2} {(5 y)}^{4} + 6 (4 x) {(5 y)}^{5} + {(5 y)}^{6}$

Raise $5$ to the power of $2$ .

$4096 x^{6} + 30720 x^{5} y + 3840 \cdot 25 x^{4} y^{2} + 20 {(4 x)}^{3} {(5 y)}^{3} + 15 {(4 x)}^{2} {(5 y)}^{4} + 6 (4 x) {(5 y)}^{5} + {(5 y)}^{6}$

Multiply $3840$ by $25$ .

$4096 x^{6} + 30720 x^{5} y + 96000 x^{4} y^{2} + 20 {(4 x)}^{3} {(5 y)}^{3} + 15 {(4 x)}^{2} {(5 y)}^{4} + 6 (4 x) {(5 y)}^{5} + {(5 y)}^{6}$

Apply the product rule to $4 x$ .

$4096 x^{6} + 30720 x^{5} y + 96000 x^{4} y^{2} + 20 (4^{3} x^{3}) {(5 y)}^{3} + 15 {(4 x)}^{2} {(5 y)}^{4} + 6 (4 x) {(5 y)}^{5} + {(5 y)}^{6}$

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

$4096 x^{6} + 30720 x^{5} y + 96000 x^{4} y^{2} + 20 (64 x^{3}) {(5 y)}^{3} + 15 {(4 x)}^{2} {(5 y)}^{4} + 6 (4 x) {(5 y)}^{5} + {(5 y)}^{6}$

Multiply $64$ by $20$ .

$4096 x^{6} + 30720 x^{5} y + 96000 x^{4} y^{2} + 1280 x^{3} {(5 y)}^{3} + 15 {(4 x)}^{2} {(5 y)}^{4} + 6 (4 x) {(5 y)}^{5} + {(5 y)}^{6}$

Apply the product rule to $5 y$ .

$4096 x^{6} + 30720 x^{5} y + 96000 x^{4} y^{2} + 1280 x^{3} (5^{3} y^{3}) + 15 {(4 x)}^{2} {(5 y)}^{4} + 6 (4 x) {(5 y)}^{5} + {(5 y)}^{6}$

Rewrite using the commutative property of multiplication.

$4096 x^{6} + 30720 x^{5} y + 96000 x^{4} y^{2} + 1280 \cdot 5^{3} x^{3} y^{3} + 15 {(4 x)}^{2} {(5 y)}^{4} + 6 (4 x) {(5 y)}^{5} + {(5 y)}^{6}$

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

$4096 x^{6} + 30720 x^{5} y + 96000 x^{4} y^{2} + 1280 \cdot 125 x^{3} y^{3} + 15 {(4 x)}^{2} {(5 y)}^{4} + 6 (4 x) {(5 y)}^{5} + {(5 y)}^{6}$

Multiply $1280$ by $125$ .

$4096 x^{6} + 30720 x^{5} y + 96000 x^{4} y^{2} + 160000 x^{3} y^{3} + 15 {(4 x)}^{2} {(5 y)}^{4} + 6 (4 x) {(5 y)}^{5} + {(5 y)}^{6}$

Apply the product rule to $4 x$ .

$4096 x^{6} + 30720 x^{5} y + 96000 x^{4} y^{2} + 160000 x^{3} y^{3} + 15 (4^{2} x^{2}) {(5 y)}^{4} + 6 (4 x) {(5 y)}^{5} + {(5 y)}^{6}$

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

$4096 x^{6} + 30720 x^{5} y + 96000 x^{4} y^{2} + 160000 x^{3} y^{3} + 15 (16 x^{2}) {(5 y)}^{4} + 6 (4 x) {(5 y)}^{5} + {(5 y)}^{6}$

Multiply $16$ by $15$ .

$4096 x^{6} + 30720 x^{5} y + 96000 x^{4} y^{2} + 160000 x^{3} y^{3} + 240 x^{2} {(5 y)}^{4} + 6 (4 x) {(5 y)}^{5} + {(5 y)}^{6}$

Apply the product rule to $5 y$ .

$4096 x^{6} + 30720 x^{5} y + 96000 x^{4} y^{2} + 160000 x^{3} y^{3} + 240 x^{2} (5^{4} y^{4}) + 6 (4 x) {(5 y)}^{5} + {(5 y)}^{6}$

Rewrite using the commutative property of multiplication.

$4096 x^{6} + 30720 x^{5} y + 96000 x^{4} y^{2} + 160000 x^{3} y^{3} + 240 \cdot 5^{4} x^{2} y^{4} + 6 (4 x) {(5 y)}^{5} + {(5 y)}^{6}$

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

$4096 x^{6} + 30720 x^{5} y + 96000 x^{4} y^{2} + 160000 x^{3} y^{3} + 240 \cdot 625 x^{2} y^{4} + 6 (4 x) {(5 y)}^{5} + {(5 y)}^{6}$

Multiply $240$ by $625$ .

$4096 x^{6} + 30720 x^{5} y + 96000 x^{4} y^{2} + 160000 x^{3} y^{3} + 150000 x^{2} y^{4} + 6 (4 x) {(5 y)}^{5} + {(5 y)}^{6}$

Multiply $4$ by $6$ .

$4096 x^{6} + 30720 x^{5} y + 96000 x^{4} y^{2} + 160000 x^{3} y^{3} + 150000 x^{2} y^{4} + 24 x {(5 y)}^{5} + {(5 y)}^{6}$

Apply the product rule to $5 y$ .

$4096 x^{6} + 30720 x^{5} y + 96000 x^{4} y^{2} + 160000 x^{3} y^{3} + 150000 x^{2} y^{4} + 24 x (5^{5} y^{5}) + {(5 y)}^{6}$

Rewrite using the commutative property of multiplication.

$4096 x^{6} + 30720 x^{5} y + 96000 x^{4} y^{2} + 160000 x^{3} y^{3} + 150000 x^{2} y^{4} + 24 \cdot 5^{5} x y^{5} + {(5 y)}^{6}$

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

$4096 x^{6} + 30720 x^{5} y + 96000 x^{4} y^{2} + 160000 x^{3} y^{3} + 150000 x^{2} y^{4} + 24 \cdot 3125 x y^{5} + {(5 y)}^{6}$

Multiply $24$ by $3125$ .

$4096 x^{6} + 30720 x^{5} y + 96000 x^{4} y^{2} + 160000 x^{3} y^{3} + 150000 x^{2} y^{4} + 75000 x y^{5} + {(5 y)}^{6}$

Apply the product rule to $5 y$ .

$4096 x^{6} + 30720 x^{5} y + 96000 x^{4} y^{2} + 160000 x^{3} y^{3} + 150000 x^{2} y^{4} + 75000 x y^{5} + 5^{6} y^{6}$

Raise $5$ to the power of $6$ .

$4096 x^{6} + 30720 x^{5} y + 96000 x^{4} y^{2} + 160000 x^{3} y^{3} + 150000 x^{2} y^{4} + 75000 x y^{5} + 15625 y^{6}$

Expand the Trigonometric Expression (4x+5y)^6

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