Expand Using Pascal's Triangle (2a-3b)^10

${(2 a - 3 b)}^{10}$

Step 1

Pascal\'s Triangle can be displayed as such:

$1$

$1 - 1$

$1 - 2 - 1$

$\dots$

$1 - 8 - 28 - 56 - 70 - 56 - 28 - 8 - 1$

$1 - 9 - 36 - 84 - 126 - 126 - 84 - 36 - 9 - 1$

$1 - 10 - 45 - 120 - 210 - 252 - 210 - 120 - 45 - 10 - 1$

The triangle can be used to calculate the coefficients of the expansion of ${(a + b)}^{n}$ by taking the exponent $n$ and adding $1$ . The coefficients will correspond with line $n + 1$ of the triangle. For ${(2 a - 3 b)}^{10}$ , $n = 10$ so the coefficients of the expansion will correspond with line $11$ .

Step 2

The expansion follows the rule ${(a + b)}^{n} = c_{0} a^{n} b^{0} + c_{1} a^{n - 1} b^{1} + c_{n - 1} a^{1} b^{n - 1} + c_{n} a^{0} b^{n}$ . The values of the coefficients, from the triangle, are $1 - 10 - 45 - 120 - 210 - 252 - 210 - 120 - 45 - 10 - 1$ .

$1 a^{10} b^{0} + 10 a^{9} b + 45 a^{8} b^{2} + 120 a^{7} b^{3} + 210 a^{6} b^{4} + 252 a^{5} b^{5} + 210 a^{4} b^{6} + 120 a^{3} b^{7} + 45 a^{2} b^{8} + 10 a b^{9} + 1 a^{0} b^{10}$

Step 3

Substitute the actual values of $a$ $2 a$ and $b$ $- 3 b$ into the expression.

$1 {(2 a)}^{10} {(- 3 b)}^{0} + 10 {(2 a)}^{9} {(- 3 b)}^{1} + 45 {(2 a)}^{8} {(- 3 b)}^{2} + 120 {(2 a)}^{7} {(- 3 b)}^{3} + 210 {(2 a)}^{6} {(- 3 b)}^{4} + 252 {(2 a)}^{5} {(- 3 b)}^{5} + 210 {(2 a)}^{4} {(- 3 b)}^{6} + 120 {(2 a)}^{3} {(- 3 b)}^{7} + 45 {(2 a)}^{2} {(- 3 b)}^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

Step 4

Simplify each term.

Multiply ${(2 a)}^{10}$ by $1$ .

${(2 a)}^{10} {(- 3 b)}^{0} + 10 {(2 a)}^{9} {(- 3 b)}^{1} + 45 {(2 a)}^{8} {(- 3 b)}^{2} + 120 {(2 a)}^{7} {(- 3 b)}^{3} + 210 {(2 a)}^{6} {(- 3 b)}^{4} + 252 {(2 a)}^{5} {(- 3 b)}^{5} + 210 {(2 a)}^{4} {(- 3 b)}^{6} + 120 {(2 a)}^{3} {(- 3 b)}^{7} + 45 {(2 a)}^{2} {(- 3 b)}^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

Apply the product rule to $2 a$ .

$2^{10} a^{10} {(- 3 b)}^{0} + 10 {(2 a)}^{9} {(- 3 b)}^{1} + 45 {(2 a)}^{8} {(- 3 b)}^{2} + 120 {(2 a)}^{7} {(- 3 b)}^{3} + 210 {(2 a)}^{6} {(- 3 b)}^{4} + 252 {(2 a)}^{5} {(- 3 b)}^{5} + 210 {(2 a)}^{4} {(- 3 b)}^{6} + 120 {(2 a)}^{3} {(- 3 b)}^{7} + 45 {(2 a)}^{2} {(- 3 b)}^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

Raise $2$ to the power of $10$ .

$1024 a^{10} {(- 3 b)}^{0} + 10 {(2 a)}^{9} {(- 3 b)}^{1} + 45 {(2 a)}^{8} {(- 3 b)}^{2} + 120 {(2 a)}^{7} {(- 3 b)}^{3} + 210 {(2 a)}^{6} {(- 3 b)}^{4} + 252 {(2 a)}^{5} {(- 3 b)}^{5} + 210 {(2 a)}^{4} {(- 3 b)}^{6} + 120 {(2 a)}^{3} {(- 3 b)}^{7} + 45 {(2 a)}^{2} {(- 3 b)}^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

Apply the product rule to $- 3 b$ .

$1024 a^{10} ({(- 3)}^{0} b^{0}) + 10 {(2 a)}^{9} {(- 3 b)}^{1} + 45 {(2 a)}^{8} {(- 3 b)}^{2} + 120 {(2 a)}^{7} {(- 3 b)}^{3} + 210 {(2 a)}^{6} {(- 3 b)}^{4} + 252 {(2 a)}^{5} {(- 3 b)}^{5} + 210 {(2 a)}^{4} {(- 3 b)}^{6} + 120 {(2 a)}^{3} {(- 3 b)}^{7} + 45 {(2 a)}^{2} {(- 3 b)}^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

Rewrite using the commutative property of multiplication.

$1024 \cdot {(- 3)}^{0} a^{10} b^{0} + 10 {(2 a)}^{9} {(- 3 b)}^{1} + 45 {(2 a)}^{8} {(- 3 b)}^{2} + 120 {(2 a)}^{7} {(- 3 b)}^{3} + 210 {(2 a)}^{6} {(- 3 b)}^{4} + 252 {(2 a)}^{5} {(- 3 b)}^{5} + 210 {(2 a)}^{4} {(- 3 b)}^{6} + 120 {(2 a)}^{3} {(- 3 b)}^{7} + 45 {(2 a)}^{2} {(- 3 b)}^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

Anything raised to $0$ is $1$ .

$1024 \cdot 1 a^{10} b^{0} + 10 {(2 a)}^{9} {(- 3 b)}^{1} + 45 {(2 a)}^{8} {(- 3 b)}^{2} + 120 {(2 a)}^{7} {(- 3 b)}^{3} + 210 {(2 a)}^{6} {(- 3 b)}^{4} + 252 {(2 a)}^{5} {(- 3 b)}^{5} + 210 {(2 a)}^{4} {(- 3 b)}^{6} + 120 {(2 a)}^{3} {(- 3 b)}^{7} + 45 {(2 a)}^{2} {(- 3 b)}^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

Multiply $1024$ by $1$ .

$1024 a^{10} b^{0} + 10 {(2 a)}^{9} {(- 3 b)}^{1} + 45 {(2 a)}^{8} {(- 3 b)}^{2} + 120 {(2 a)}^{7} {(- 3 b)}^{3} + 210 {(2 a)}^{6} {(- 3 b)}^{4} + 252 {(2 a)}^{5} {(- 3 b)}^{5} + 210 {(2 a)}^{4} {(- 3 b)}^{6} + 120 {(2 a)}^{3} {(- 3 b)}^{7} + 45 {(2 a)}^{2} {(- 3 b)}^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

Anything raised to $0$ is $1$ .

$1024 a^{10} \cdot 1 + 10 {(2 a)}^{9} {(- 3 b)}^{1} + 45 {(2 a)}^{8} {(- 3 b)}^{2} + 120 {(2 a)}^{7} {(- 3 b)}^{3} + 210 {(2 a)}^{6} {(- 3 b)}^{4} + 252 {(2 a)}^{5} {(- 3 b)}^{5} + 210 {(2 a)}^{4} {(- 3 b)}^{6} + 120 {(2 a)}^{3} {(- 3 b)}^{7} + 45 {(2 a)}^{2} {(- 3 b)}^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

Multiply $1024$ by $1$ .

$1024 a^{10} + 10 {(2 a)}^{9} {(- 3 b)}^{1} + 45 {(2 a)}^{8} {(- 3 b)}^{2} + 120 {(2 a)}^{7} {(- 3 b)}^{3} + 210 {(2 a)}^{6} {(- 3 b)}^{4} + 252 {(2 a)}^{5} {(- 3 b)}^{5} + 210 {(2 a)}^{4} {(- 3 b)}^{6} + 120 {(2 a)}^{3} {(- 3 b)}^{7} + 45 {(2 a)}^{2} {(- 3 b)}^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

Apply the product rule to $2 a$ .

$1024 a^{10} + 10 (2^{9} a^{9}) {(- 3 b)}^{1} + 45 {(2 a)}^{8} {(- 3 b)}^{2} + 120 {(2 a)}^{7} {(- 3 b)}^{3} + 210 {(2 a)}^{6} {(- 3 b)}^{4} + 252 {(2 a)}^{5} {(- 3 b)}^{5} + 210 {(2 a)}^{4} {(- 3 b)}^{6} + 120 {(2 a)}^{3} {(- 3 b)}^{7} + 45 {(2 a)}^{2} {(- 3 b)}^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

Raise $2$ to the power of $9$ .

$1024 a^{10} + 10 (512 a^{9}) {(- 3 b)}^{1} + 45 {(2 a)}^{8} {(- 3 b)}^{2} + 120 {(2 a)}^{7} {(- 3 b)}^{3} + 210 {(2 a)}^{6} {(- 3 b)}^{4} + 252 {(2 a)}^{5} {(- 3 b)}^{5} + 210 {(2 a)}^{4} {(- 3 b)}^{6} + 120 {(2 a)}^{3} {(- 3 b)}^{7} + 45 {(2 a)}^{2} {(- 3 b)}^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

Multiply $512$ by $10$ .

$1024 a^{10} + 5120 a^{9} {(- 3 b)}^{1} + 45 {(2 a)}^{8} {(- 3 b)}^{2} + 120 {(2 a)}^{7} {(- 3 b)}^{3} + 210 {(2 a)}^{6} {(- 3 b)}^{4} + 252 {(2 a)}^{5} {(- 3 b)}^{5} + 210 {(2 a)}^{4} {(- 3 b)}^{6} + 120 {(2 a)}^{3} {(- 3 b)}^{7} + 45 {(2 a)}^{2} {(- 3 b)}^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

Simplify.

$1024 a^{10} + 5120 a^{9} (- 3 b) + 45 {(2 a)}^{8} {(- 3 b)}^{2} + 120 {(2 a)}^{7} {(- 3 b)}^{3} + 210 {(2 a)}^{6} {(- 3 b)}^{4} + 252 {(2 a)}^{5} {(- 3 b)}^{5} + 210 {(2 a)}^{4} {(- 3 b)}^{6} + 120 {(2 a)}^{3} {(- 3 b)}^{7} + 45 {(2 a)}^{2} {(- 3 b)}^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

Rewrite using the commutative property of multiplication.

$1024 a^{10} + 5120 \cdot - 3 a^{9} b + 45 {(2 a)}^{8} {(- 3 b)}^{2} + 120 {(2 a)}^{7} {(- 3 b)}^{3} + 210 {(2 a)}^{6} {(- 3 b)}^{4} + 252 {(2 a)}^{5} {(- 3 b)}^{5} + 210 {(2 a)}^{4} {(- 3 b)}^{6} + 120 {(2 a)}^{3} {(- 3 b)}^{7} + 45 {(2 a)}^{2} {(- 3 b)}^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

Multiply $5120$ by $- 3$ .

$1024 a^{10} - 15360 a^{9} b + 45 {(2 a)}^{8} {(- 3 b)}^{2} + 120 {(2 a)}^{7} {(- 3 b)}^{3} + 210 {(2 a)}^{6} {(- 3 b)}^{4} + 252 {(2 a)}^{5} {(- 3 b)}^{5} + 210 {(2 a)}^{4} {(- 3 b)}^{6} + 120 {(2 a)}^{3} {(- 3 b)}^{7} + 45 {(2 a)}^{2} {(- 3 b)}^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

Apply the product rule to $2 a$ .

$1024 a^{10} - 15360 a^{9} b + 45 (2^{8} a^{8}) {(- 3 b)}^{2} + 120 {(2 a)}^{7} {(- 3 b)}^{3} + 210 {(2 a)}^{6} {(- 3 b)}^{4} + 252 {(2 a)}^{5} {(- 3 b)}^{5} + 210 {(2 a)}^{4} {(- 3 b)}^{6} + 120 {(2 a)}^{3} {(- 3 b)}^{7} + 45 {(2 a)}^{2} {(- 3 b)}^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

Raise $2$ to the power of $8$ .

$1024 a^{10} - 15360 a^{9} b + 45 (256 a^{8}) {(- 3 b)}^{2} + 120 {(2 a)}^{7} {(- 3 b)}^{3} + 210 {(2 a)}^{6} {(- 3 b)}^{4} + 252 {(2 a)}^{5} {(- 3 b)}^{5} + 210 {(2 a)}^{4} {(- 3 b)}^{6} + 120 {(2 a)}^{3} {(- 3 b)}^{7} + 45 {(2 a)}^{2} {(- 3 b)}^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

Multiply $256$ by $45$ .

$1024 a^{10} - 15360 a^{9} b + 11520 a^{8} {(- 3 b)}^{2} + 120 {(2 a)}^{7} {(- 3 b)}^{3} + 210 {(2 a)}^{6} {(- 3 b)}^{4} + 252 {(2 a)}^{5} {(- 3 b)}^{5} + 210 {(2 a)}^{4} {(- 3 b)}^{6} + 120 {(2 a)}^{3} {(- 3 b)}^{7} + 45 {(2 a)}^{2} {(- 3 b)}^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

Apply the product rule to $- 3 b$ .

$1024 a^{10} - 15360 a^{9} b + 11520 a^{8} ({(- 3)}^{2} b^{2}) + 120 {(2 a)}^{7} {(- 3 b)}^{3} + 210 {(2 a)}^{6} {(- 3 b)}^{4} + 252 {(2 a)}^{5} {(- 3 b)}^{5} + 210 {(2 a)}^{4} {(- 3 b)}^{6} + 120 {(2 a)}^{3} {(- 3 b)}^{7} + 45 {(2 a)}^{2} {(- 3 b)}^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

Rewrite using the commutative property of multiplication.

$1024 a^{10} - 15360 a^{9} b + 11520 \cdot {(- 3)}^{2} a^{8} b^{2} + 120 {(2 a)}^{7} {(- 3 b)}^{3} + 210 {(2 a)}^{6} {(- 3 b)}^{4} + 252 {(2 a)}^{5} {(- 3 b)}^{5} + 210 {(2 a)}^{4} {(- 3 b)}^{6} + 120 {(2 a)}^{3} {(- 3 b)}^{7} + 45 {(2 a)}^{2} {(- 3 b)}^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

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

$1024 a^{10} - 15360 a^{9} b + 11520 \cdot 9 a^{8} b^{2} + 120 {(2 a)}^{7} {(- 3 b)}^{3} + 210 {(2 a)}^{6} {(- 3 b)}^{4} + 252 {(2 a)}^{5} {(- 3 b)}^{5} + 210 {(2 a)}^{4} {(- 3 b)}^{6} + 120 {(2 a)}^{3} {(- 3 b)}^{7} + 45 {(2 a)}^{2} {(- 3 b)}^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

Multiply $11520$ by $9$ .

$1024 a^{10} - 15360 a^{9} b + 103680 a^{8} b^{2} + 120 {(2 a)}^{7} {(- 3 b)}^{3} + 210 {(2 a)}^{6} {(- 3 b)}^{4} + 252 {(2 a)}^{5} {(- 3 b)}^{5} + 210 {(2 a)}^{4} {(- 3 b)}^{6} + 120 {(2 a)}^{3} {(- 3 b)}^{7} + 45 {(2 a)}^{2} {(- 3 b)}^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

Apply the product rule to $2 a$ .

$1024 a^{10} - 15360 a^{9} b + 103680 a^{8} b^{2} + 120 (2^{7} a^{7}) {(- 3 b)}^{3} + 210 {(2 a)}^{6} {(- 3 b)}^{4} + 252 {(2 a)}^{5} {(- 3 b)}^{5} + 210 {(2 a)}^{4} {(- 3 b)}^{6} + 120 {(2 a)}^{3} {(- 3 b)}^{7} + 45 {(2 a)}^{2} {(- 3 b)}^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

Raise $2$ to the power of $7$ .

$1024 a^{10} - 15360 a^{9} b + 103680 a^{8} b^{2} + 120 (128 a^{7}) {(- 3 b)}^{3} + 210 {(2 a)}^{6} {(- 3 b)}^{4} + 252 {(2 a)}^{5} {(- 3 b)}^{5} + 210 {(2 a)}^{4} {(- 3 b)}^{6} + 120 {(2 a)}^{3} {(- 3 b)}^{7} + 45 {(2 a)}^{2} {(- 3 b)}^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

Multiply $128$ by $120$ .

$1024 a^{10} - 15360 a^{9} b + 103680 a^{8} b^{2} + 15360 a^{7} {(- 3 b)}^{3} + 210 {(2 a)}^{6} {(- 3 b)}^{4} + 252 {(2 a)}^{5} {(- 3 b)}^{5} + 210 {(2 a)}^{4} {(- 3 b)}^{6} + 120 {(2 a)}^{3} {(- 3 b)}^{7} + 45 {(2 a)}^{2} {(- 3 b)}^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

Apply the product rule to $- 3 b$ .

$1024 a^{10} - 15360 a^{9} b + 103680 a^{8} b^{2} + 15360 a^{7} ({(- 3)}^{3} b^{3}) + 210 {(2 a)}^{6} {(- 3 b)}^{4} + 252 {(2 a)}^{5} {(- 3 b)}^{5} + 210 {(2 a)}^{4} {(- 3 b)}^{6} + 120 {(2 a)}^{3} {(- 3 b)}^{7} + 45 {(2 a)}^{2} {(- 3 b)}^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

Rewrite using the commutative property of multiplication.

$1024 a^{10} - 15360 a^{9} b + 103680 a^{8} b^{2} + 15360 \cdot {(- 3)}^{3} a^{7} b^{3} + 210 {(2 a)}^{6} {(- 3 b)}^{4} + 252 {(2 a)}^{5} {(- 3 b)}^{5} + 210 {(2 a)}^{4} {(- 3 b)}^{6} + 120 {(2 a)}^{3} {(- 3 b)}^{7} + 45 {(2 a)}^{2} {(- 3 b)}^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

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

$1024 a^{10} - 15360 a^{9} b + 103680 a^{8} b^{2} + 15360 \cdot - 27 a^{7} b^{3} + 210 {(2 a)}^{6} {(- 3 b)}^{4} + 252 {(2 a)}^{5} {(- 3 b)}^{5} + 210 {(2 a)}^{4} {(- 3 b)}^{6} + 120 {(2 a)}^{3} {(- 3 b)}^{7} + 45 {(2 a)}^{2} {(- 3 b)}^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

Multiply $15360$ by $- 27$ .

$1024 a^{10} - 15360 a^{9} b + 103680 a^{8} b^{2} - 414720 a^{7} b^{3} + 210 {(2 a)}^{6} {(- 3 b)}^{4} + 252 {(2 a)}^{5} {(- 3 b)}^{5} + 210 {(2 a)}^{4} {(- 3 b)}^{6} + 120 {(2 a)}^{3} {(- 3 b)}^{7} + 45 {(2 a)}^{2} {(- 3 b)}^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

Apply the product rule to $2 a$ .

$1024 a^{10} - 15360 a^{9} b + 103680 a^{8} b^{2} - 414720 a^{7} b^{3} + 210 (2^{6} a^{6}) {(- 3 b)}^{4} + 252 {(2 a)}^{5} {(- 3 b)}^{5} + 210 {(2 a)}^{4} {(- 3 b)}^{6} + 120 {(2 a)}^{3} {(- 3 b)}^{7} + 45 {(2 a)}^{2} {(- 3 b)}^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

Raise $2$ to the power of $6$ .

$1024 a^{10} - 15360 a^{9} b + 103680 a^{8} b^{2} - 414720 a^{7} b^{3} + 210 (64 a^{6}) {(- 3 b)}^{4} + 252 {(2 a)}^{5} {(- 3 b)}^{5} + 210 {(2 a)}^{4} {(- 3 b)}^{6} + 120 {(2 a)}^{3} {(- 3 b)}^{7} + 45 {(2 a)}^{2} {(- 3 b)}^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

Multiply $64$ by $210$ .

$1024 a^{10} - 15360 a^{9} b + 103680 a^{8} b^{2} - 414720 a^{7} b^{3} + 13440 a^{6} {(- 3 b)}^{4} + 252 {(2 a)}^{5} {(- 3 b)}^{5} + 210 {(2 a)}^{4} {(- 3 b)}^{6} + 120 {(2 a)}^{3} {(- 3 b)}^{7} + 45 {(2 a)}^{2} {(- 3 b)}^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

Apply the product rule to $- 3 b$ .

$1024 a^{10} - 15360 a^{9} b + 103680 a^{8} b^{2} - 414720 a^{7} b^{3} + 13440 a^{6} ({(- 3)}^{4} b^{4}) + 252 {(2 a)}^{5} {(- 3 b)}^{5} + 210 {(2 a)}^{4} {(- 3 b)}^{6} + 120 {(2 a)}^{3} {(- 3 b)}^{7} + 45 {(2 a)}^{2} {(- 3 b)}^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

Rewrite using the commutative property of multiplication.

$1024 a^{10} - 15360 a^{9} b + 103680 a^{8} b^{2} - 414720 a^{7} b^{3} + 13440 \cdot {(- 3)}^{4} a^{6} b^{4} + 252 {(2 a)}^{5} {(- 3 b)}^{5} + 210 {(2 a)}^{4} {(- 3 b)}^{6} + 120 {(2 a)}^{3} {(- 3 b)}^{7} + 45 {(2 a)}^{2} {(- 3 b)}^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

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

$1024 a^{10} - 15360 a^{9} b + 103680 a^{8} b^{2} - 414720 a^{7} b^{3} + 13440 \cdot 81 a^{6} b^{4} + 252 {(2 a)}^{5} {(- 3 b)}^{5} + 210 {(2 a)}^{4} {(- 3 b)}^{6} + 120 {(2 a)}^{3} {(- 3 b)}^{7} + 45 {(2 a)}^{2} {(- 3 b)}^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

Multiply $13440$ by $81$ .

$1024 a^{10} - 15360 a^{9} b + 103680 a^{8} b^{2} - 414720 a^{7} b^{3} + 1088640 a^{6} b^{4} + 252 {(2 a)}^{5} {(- 3 b)}^{5} + 210 {(2 a)}^{4} {(- 3 b)}^{6} + 120 {(2 a)}^{3} {(- 3 b)}^{7} + 45 {(2 a)}^{2} {(- 3 b)}^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

Apply the product rule to $2 a$ .

$1024 a^{10} - 15360 a^{9} b + 103680 a^{8} b^{2} - 414720 a^{7} b^{3} + 1088640 a^{6} b^{4} + 252 (2^{5} a^{5}) {(- 3 b)}^{5} + 210 {(2 a)}^{4} {(- 3 b)}^{6} + 120 {(2 a)}^{3} {(- 3 b)}^{7} + 45 {(2 a)}^{2} {(- 3 b)}^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

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

$1024 a^{10} - 15360 a^{9} b + 103680 a^{8} b^{2} - 414720 a^{7} b^{3} + 1088640 a^{6} b^{4} + 252 (32 a^{5}) {(- 3 b)}^{5} + 210 {(2 a)}^{4} {(- 3 b)}^{6} + 120 {(2 a)}^{3} {(- 3 b)}^{7} + 45 {(2 a)}^{2} {(- 3 b)}^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

Multiply $32$ by $252$ .

$1024 a^{10} - 15360 a^{9} b + 103680 a^{8} b^{2} - 414720 a^{7} b^{3} + 1088640 a^{6} b^{4} + 8064 a^{5} {(- 3 b)}^{5} + 210 {(2 a)}^{4} {(- 3 b)}^{6} + 120 {(2 a)}^{3} {(- 3 b)}^{7} + 45 {(2 a)}^{2} {(- 3 b)}^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

Apply the product rule to $- 3 b$ .

$1024 a^{10} - 15360 a^{9} b + 103680 a^{8} b^{2} - 414720 a^{7} b^{3} + 1088640 a^{6} b^{4} + 8064 a^{5} ({(- 3)}^{5} b^{5}) + 210 {(2 a)}^{4} {(- 3 b)}^{6} + 120 {(2 a)}^{3} {(- 3 b)}^{7} + 45 {(2 a)}^{2} {(- 3 b)}^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

Rewrite using the commutative property of multiplication.

$1024 a^{10} - 15360 a^{9} b + 103680 a^{8} b^{2} - 414720 a^{7} b^{3} + 1088640 a^{6} b^{4} + 8064 \cdot {(- 3)}^{5} a^{5} b^{5} + 210 {(2 a)}^{4} {(- 3 b)}^{6} + 120 {(2 a)}^{3} {(- 3 b)}^{7} + 45 {(2 a)}^{2} {(- 3 b)}^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

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

$1024 a^{10} - 15360 a^{9} b + 103680 a^{8} b^{2} - 414720 a^{7} b^{3} + 1088640 a^{6} b^{4} + 8064 \cdot - 243 a^{5} b^{5} + 210 {(2 a)}^{4} {(- 3 b)}^{6} + 120 {(2 a)}^{3} {(- 3 b)}^{7} + 45 {(2 a)}^{2} {(- 3 b)}^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

Multiply $8064$ by $- 243$ .

$1024 a^{10} - 15360 a^{9} b + 103680 a^{8} b^{2} - 414720 a^{7} b^{3} + 1088640 a^{6} b^{4} - 1959552 a^{5} b^{5} + 210 {(2 a)}^{4} {(- 3 b)}^{6} + 120 {(2 a)}^{3} {(- 3 b)}^{7} + 45 {(2 a)}^{2} {(- 3 b)}^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

Apply the product rule to $2 a$ .

$1024 a^{10} - 15360 a^{9} b + 103680 a^{8} b^{2} - 414720 a^{7} b^{3} + 1088640 a^{6} b^{4} - 1959552 a^{5} b^{5} + 210 (2^{4} a^{4}) {(- 3 b)}^{6} + 120 {(2 a)}^{3} {(- 3 b)}^{7} + 45 {(2 a)}^{2} {(- 3 b)}^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

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

$1024 a^{10} - 15360 a^{9} b + 103680 a^{8} b^{2} - 414720 a^{7} b^{3} + 1088640 a^{6} b^{4} - 1959552 a^{5} b^{5} + 210 (16 a^{4}) {(- 3 b)}^{6} + 120 {(2 a)}^{3} {(- 3 b)}^{7} + 45 {(2 a)}^{2} {(- 3 b)}^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

Multiply $16$ by $210$ .

$1024 a^{10} - 15360 a^{9} b + 103680 a^{8} b^{2} - 414720 a^{7} b^{3} + 1088640 a^{6} b^{4} - 1959552 a^{5} b^{5} + 3360 a^{4} {(- 3 b)}^{6} + 120 {(2 a)}^{3} {(- 3 b)}^{7} + 45 {(2 a)}^{2} {(- 3 b)}^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

Apply the product rule to $- 3 b$ .

$1024 a^{10} - 15360 a^{9} b + 103680 a^{8} b^{2} - 414720 a^{7} b^{3} + 1088640 a^{6} b^{4} - 1959552 a^{5} b^{5} + 3360 a^{4} ({(- 3)}^{6} b^{6}) + 120 {(2 a)}^{3} {(- 3 b)}^{7} + 45 {(2 a)}^{2} {(- 3 b)}^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

Rewrite using the commutative property of multiplication.

$1024 a^{10} - 15360 a^{9} b + 103680 a^{8} b^{2} - 414720 a^{7} b^{3} + 1088640 a^{6} b^{4} - 1959552 a^{5} b^{5} + 3360 \cdot {(- 3)}^{6} a^{4} b^{6} + 120 {(2 a)}^{3} {(- 3 b)}^{7} + 45 {(2 a)}^{2} {(- 3 b)}^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

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

$1024 a^{10} - 15360 a^{9} b + 103680 a^{8} b^{2} - 414720 a^{7} b^{3} + 1088640 a^{6} b^{4} - 1959552 a^{5} b^{5} + 3360 \cdot 729 a^{4} b^{6} + 120 {(2 a)}^{3} {(- 3 b)}^{7} + 45 {(2 a)}^{2} {(- 3 b)}^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

Multiply $3360$ by $729$ .

$1024 a^{10} - 15360 a^{9} b + 103680 a^{8} b^{2} - 414720 a^{7} b^{3} + 1088640 a^{6} b^{4} - 1959552 a^{5} b^{5} + 2449440 a^{4} b^{6} + 120 {(2 a)}^{3} {(- 3 b)}^{7} + 45 {(2 a)}^{2} {(- 3 b)}^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

Apply the product rule to $2 a$ .

$1024 a^{10} - 15360 a^{9} b + 103680 a^{8} b^{2} - 414720 a^{7} b^{3} + 1088640 a^{6} b^{4} - 1959552 a^{5} b^{5} + 2449440 a^{4} b^{6} + 120 (2^{3} a^{3}) {(- 3 b)}^{7} + 45 {(2 a)}^{2} {(- 3 b)}^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

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

$1024 a^{10} - 15360 a^{9} b + 103680 a^{8} b^{2} - 414720 a^{7} b^{3} + 1088640 a^{6} b^{4} - 1959552 a^{5} b^{5} + 2449440 a^{4} b^{6} + 120 (8 a^{3}) {(- 3 b)}^{7} + 45 {(2 a)}^{2} {(- 3 b)}^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

Multiply $8$ by $120$ .

$1024 a^{10} - 15360 a^{9} b + 103680 a^{8} b^{2} - 414720 a^{7} b^{3} + 1088640 a^{6} b^{4} - 1959552 a^{5} b^{5} + 2449440 a^{4} b^{6} + 960 a^{3} {(- 3 b)}^{7} + 45 {(2 a)}^{2} {(- 3 b)}^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

Apply the product rule to $- 3 b$ .

$1024 a^{10} - 15360 a^{9} b + 103680 a^{8} b^{2} - 414720 a^{7} b^{3} + 1088640 a^{6} b^{4} - 1959552 a^{5} b^{5} + 2449440 a^{4} b^{6} + 960 a^{3} ({(- 3)}^{7} b^{7}) + 45 {(2 a)}^{2} {(- 3 b)}^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

Rewrite using the commutative property of multiplication.

$1024 a^{10} - 15360 a^{9} b + 103680 a^{8} b^{2} - 414720 a^{7} b^{3} + 1088640 a^{6} b^{4} - 1959552 a^{5} b^{5} + 2449440 a^{4} b^{6} + 960 \cdot {(- 3)}^{7} a^{3} b^{7} + 45 {(2 a)}^{2} {(- 3 b)}^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

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

$1024 a^{10} - 15360 a^{9} b + 103680 a^{8} b^{2} - 414720 a^{7} b^{3} + 1088640 a^{6} b^{4} - 1959552 a^{5} b^{5} + 2449440 a^{4} b^{6} + 960 \cdot - 2187 a^{3} b^{7} + 45 {(2 a)}^{2} {(- 3 b)}^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

Multiply $960$ by $- 2187$ .

$1024 a^{10} - 15360 a^{9} b + 103680 a^{8} b^{2} - 414720 a^{7} b^{3} + 1088640 a^{6} b^{4} - 1959552 a^{5} b^{5} + 2449440 a^{4} b^{6} - 2099520 a^{3} b^{7} + 45 {(2 a)}^{2} {(- 3 b)}^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

Apply the product rule to $2 a$ .

$1024 a^{10} - 15360 a^{9} b + 103680 a^{8} b^{2} - 414720 a^{7} b^{3} + 1088640 a^{6} b^{4} - 1959552 a^{5} b^{5} + 2449440 a^{4} b^{6} - 2099520 a^{3} b^{7} + 45 (2^{2} a^{2}) {(- 3 b)}^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

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

$1024 a^{10} - 15360 a^{9} b + 103680 a^{8} b^{2} - 414720 a^{7} b^{3} + 1088640 a^{6} b^{4} - 1959552 a^{5} b^{5} + 2449440 a^{4} b^{6} - 2099520 a^{3} b^{7} + 45 (4 a^{2}) {(- 3 b)}^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

Multiply $4$ by $45$ .

$1024 a^{10} - 15360 a^{9} b + 103680 a^{8} b^{2} - 414720 a^{7} b^{3} + 1088640 a^{6} b^{4} - 1959552 a^{5} b^{5} + 2449440 a^{4} b^{6} - 2099520 a^{3} b^{7} + 180 a^{2} {(- 3 b)}^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

Apply the product rule to $- 3 b$ .

$1024 a^{10} - 15360 a^{9} b + 103680 a^{8} b^{2} - 414720 a^{7} b^{3} + 1088640 a^{6} b^{4} - 1959552 a^{5} b^{5} + 2449440 a^{4} b^{6} - 2099520 a^{3} b^{7} + 180 a^{2} ({(- 3)}^{8} b^{8}) + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

Rewrite using the commutative property of multiplication.

$1024 a^{10} - 15360 a^{9} b + 103680 a^{8} b^{2} - 414720 a^{7} b^{3} + 1088640 a^{6} b^{4} - 1959552 a^{5} b^{5} + 2449440 a^{4} b^{6} - 2099520 a^{3} b^{7} + 180 \cdot {(- 3)}^{8} a^{2} b^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

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

$1024 a^{10} - 15360 a^{9} b + 103680 a^{8} b^{2} - 414720 a^{7} b^{3} + 1088640 a^{6} b^{4} - 1959552 a^{5} b^{5} + 2449440 a^{4} b^{6} - 2099520 a^{3} b^{7} + 180 \cdot 6561 a^{2} b^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

Multiply $180$ by $6561$ .

$1024 a^{10} - 15360 a^{9} b + 103680 a^{8} b^{2} - 414720 a^{7} b^{3} + 1088640 a^{6} b^{4} - 1959552 a^{5} b^{5} + 2449440 a^{4} b^{6} - 2099520 a^{3} b^{7} + 1180980 a^{2} b^{8} + 10 {(2 a)}^{1} {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

Simplify.

$1024 a^{10} - 15360 a^{9} b + 103680 a^{8} b^{2} - 414720 a^{7} b^{3} + 1088640 a^{6} b^{4} - 1959552 a^{5} b^{5} + 2449440 a^{4} b^{6} - 2099520 a^{3} b^{7} + 1180980 a^{2} b^{8} + 10 (2 a) {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

Multiply $2$ by $10$ .

$1024 a^{10} - 15360 a^{9} b + 103680 a^{8} b^{2} - 414720 a^{7} b^{3} + 1088640 a^{6} b^{4} - 1959552 a^{5} b^{5} + 2449440 a^{4} b^{6} - 2099520 a^{3} b^{7} + 1180980 a^{2} b^{8} + 20 a {(- 3 b)}^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

Apply the product rule to $- 3 b$ .

Rewrite using the commutative property of multiplication.

Raise $- 3$ to the power of $9$ .

Multiply $20$ by $- 19683$ .

$1024 a^{10} - 15360 a^{9} b + 103680 a^{8} b^{2} - 414720 a^{7} b^{3} + 1088640 a^{6} b^{4} - 1959552 a^{5} b^{5} + 2449440 a^{4} b^{6} - 2099520 a^{3} b^{7} + 1180980 a^{2} b^{8} - 393660 a b^{9} + 1 {(2 a)}^{0} {(- 3 b)}^{10}$

Multiply ${(2 a)}^{0}$ by $1$ .

Apply the product rule to $2 a$ .

Anything raised to $0$ is $1$ .

Multiply $a^{0}$ by $1$ .

Anything raised to $0$ is $1$ .

Multiply ${(- 3 b)}^{10}$ by $1$ .

Apply the product rule to $- 3 b$ .

Raise $- 3$ to the power of $10$ .

Expand Using Pascal's Triangle (2a-3b)^10

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