Expand Using Pascal's Triangle (2x^3+3y^2)^7

${(2 x^{3} + 3 y^{2})}^{7}$

Step 1

Pascal\'s Triangle can be displayed as such:

$1$

$1 - 1$

$1 - 2 - 1$

$\dots$

$1 - 5 - 10 - 10 - 5 - 1$

$1 - 6 - 15 - 20 - 15 - 6 - 1$

$1 - 7 - 21 - 35 - 35 - 21 - 7 - 1$

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 ${(2 x^{3} + 3 y^{2})}^{7}$ , $n = 7$ so the coefficients of the expansion will correspond with line $8$ .

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 $1 - 7 - 21 - 35 - 35 - 21 - 7 - 1$ .

$1 a^{7} b^{0} + 7 a^{6} b + 21 a^{5} b^{2} + 35 a^{4} b^{3} + 35 a^{3} b^{4} + 21 a^{2} b^{5} + 7 a b^{6} + 1 a^{0} b^{7}$

Step 3

Substitute the actual values of $a$ $2 x^{3}$ and $b$ $3 y^{2}$ into the expression.

$1 {(2 x^{3})}^{7} {(3 y^{2})}^{0} + 7 {(2 x^{3})}^{6} {(3 y^{2})}^{1} + 21 {(2 x^{3})}^{5} {(3 y^{2})}^{2} + 35 {(2 x^{3})}^{4} {(3 y^{2})}^{3} + 35 {(2 x^{3})}^{3} {(3 y^{2})}^{4} + 21 {(2 x^{3})}^{2} {(3 y^{2})}^{5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Step 4

Simplify each term.

Multiply ${(2 x^{3})}^{7}$ by $1$ .

${(2 x^{3})}^{7} {(3 y^{2})}^{0} + 7 {(2 x^{3})}^{6} {(3 y^{2})}^{1} + 21 {(2 x^{3})}^{5} {(3 y^{2})}^{2} + 35 {(2 x^{3})}^{4} {(3 y^{2})}^{3} + 35 {(2 x^{3})}^{3} {(3 y^{2})}^{4} + 21 {(2 x^{3})}^{2} {(3 y^{2})}^{5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Apply the product rule to $2 x^{3}$ .

$2^{7} {(x^{3})}^{7} {(3 y^{2})}^{0} + 7 {(2 x^{3})}^{6} {(3 y^{2})}^{1} + 21 {(2 x^{3})}^{5} {(3 y^{2})}^{2} + 35 {(2 x^{3})}^{4} {(3 y^{2})}^{3} + 35 {(2 x^{3})}^{3} {(3 y^{2})}^{4} + 21 {(2 x^{3})}^{2} {(3 y^{2})}^{5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

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

$128 {(x^{3})}^{7} {(3 y^{2})}^{0} + 7 {(2 x^{3})}^{6} {(3 y^{2})}^{1} + 21 {(2 x^{3})}^{5} {(3 y^{2})}^{2} + 35 {(2 x^{3})}^{4} {(3 y^{2})}^{3} + 35 {(2 x^{3})}^{3} {(3 y^{2})}^{4} + 21 {(2 x^{3})}^{2} {(3 y^{2})}^{5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Multiply the exponents in ${(x^{3})}^{7}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$128 x^{3 \cdot 7} {(3 y^{2})}^{0} + 7 {(2 x^{3})}^{6} {(3 y^{2})}^{1} + 21 {(2 x^{3})}^{5} {(3 y^{2})}^{2} + 35 {(2 x^{3})}^{4} {(3 y^{2})}^{3} + 35 {(2 x^{3})}^{3} {(3 y^{2})}^{4} + 21 {(2 x^{3})}^{2} {(3 y^{2})}^{5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Multiply $3$ by $7$ .

$128 x^{21} {(3 y^{2})}^{0} + 7 {(2 x^{3})}^{6} {(3 y^{2})}^{1} + 21 {(2 x^{3})}^{5} {(3 y^{2})}^{2} + 35 {(2 x^{3})}^{4} {(3 y^{2})}^{3} + 35 {(2 x^{3})}^{3} {(3 y^{2})}^{4} + 21 {(2 x^{3})}^{2} {(3 y^{2})}^{5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Apply the product rule to $3 y^{2}$ .

$128 x^{21} (3^{0} {(y^{2})}^{0}) + 7 {(2 x^{3})}^{6} {(3 y^{2})}^{1} + 21 {(2 x^{3})}^{5} {(3 y^{2})}^{2} + 35 {(2 x^{3})}^{4} {(3 y^{2})}^{3} + 35 {(2 x^{3})}^{3} {(3 y^{2})}^{4} + 21 {(2 x^{3})}^{2} {(3 y^{2})}^{5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Rewrite using the commutative property of multiplication.

$128 \cdot 3^{0} x^{21} {(y^{2})}^{0} + 7 {(2 x^{3})}^{6} {(3 y^{2})}^{1} + 21 {(2 x^{3})}^{5} {(3 y^{2})}^{2} + 35 {(2 x^{3})}^{4} {(3 y^{2})}^{3} + 35 {(2 x^{3})}^{3} {(3 y^{2})}^{4} + 21 {(2 x^{3})}^{2} {(3 y^{2})}^{5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Anything raised to $0$ is $1$ .

$128 \cdot 1 x^{21} {(y^{2})}^{0} + 7 {(2 x^{3})}^{6} {(3 y^{2})}^{1} + 21 {(2 x^{3})}^{5} {(3 y^{2})}^{2} + 35 {(2 x^{3})}^{4} {(3 y^{2})}^{3} + 35 {(2 x^{3})}^{3} {(3 y^{2})}^{4} + 21 {(2 x^{3})}^{2} {(3 y^{2})}^{5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Multiply $128$ by $1$ .

$128 x^{21} {(y^{2})}^{0} + 7 {(2 x^{3})}^{6} {(3 y^{2})}^{1} + 21 {(2 x^{3})}^{5} {(3 y^{2})}^{2} + 35 {(2 x^{3})}^{4} {(3 y^{2})}^{3} + 35 {(2 x^{3})}^{3} {(3 y^{2})}^{4} + 21 {(2 x^{3})}^{2} {(3 y^{2})}^{5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Multiply the exponents in ${(y^{2})}^{0}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$128 x^{21} y^{2 \cdot 0} + 7 {(2 x^{3})}^{6} {(3 y^{2})}^{1} + 21 {(2 x^{3})}^{5} {(3 y^{2})}^{2} + 35 {(2 x^{3})}^{4} {(3 y^{2})}^{3} + 35 {(2 x^{3})}^{3} {(3 y^{2})}^{4} + 21 {(2 x^{3})}^{2} {(3 y^{2})}^{5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Multiply $2$ by $0$ .

$128 x^{21} y^{0} + 7 {(2 x^{3})}^{6} {(3 y^{2})}^{1} + 21 {(2 x^{3})}^{5} {(3 y^{2})}^{2} + 35 {(2 x^{3})}^{4} {(3 y^{2})}^{3} + 35 {(2 x^{3})}^{3} {(3 y^{2})}^{4} + 21 {(2 x^{3})}^{2} {(3 y^{2})}^{5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Anything raised to $0$ is $1$ .

$128 x^{21} \cdot 1 + 7 {(2 x^{3})}^{6} {(3 y^{2})}^{1} + 21 {(2 x^{3})}^{5} {(3 y^{2})}^{2} + 35 {(2 x^{3})}^{4} {(3 y^{2})}^{3} + 35 {(2 x^{3})}^{3} {(3 y^{2})}^{4} + 21 {(2 x^{3})}^{2} {(3 y^{2})}^{5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Multiply $128$ by $1$ .

$128 x^{21} + 7 {(2 x^{3})}^{6} {(3 y^{2})}^{1} + 21 {(2 x^{3})}^{5} {(3 y^{2})}^{2} + 35 {(2 x^{3})}^{4} {(3 y^{2})}^{3} + 35 {(2 x^{3})}^{3} {(3 y^{2})}^{4} + 21 {(2 x^{3})}^{2} {(3 y^{2})}^{5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Apply the product rule to $2 x^{3}$ .

$128 x^{21} + 7 (2^{6} {(x^{3})}^{6}) {(3 y^{2})}^{1} + 21 {(2 x^{3})}^{5} {(3 y^{2})}^{2} + 35 {(2 x^{3})}^{4} {(3 y^{2})}^{3} + 35 {(2 x^{3})}^{3} {(3 y^{2})}^{4} + 21 {(2 x^{3})}^{2} {(3 y^{2})}^{5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

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

$128 x^{21} + 7 (64 {(x^{3})}^{6}) {(3 y^{2})}^{1} + 21 {(2 x^{3})}^{5} {(3 y^{2})}^{2} + 35 {(2 x^{3})}^{4} {(3 y^{2})}^{3} + 35 {(2 x^{3})}^{3} {(3 y^{2})}^{4} + 21 {(2 x^{3})}^{2} {(3 y^{2})}^{5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Multiply the exponents in ${(x^{3})}^{6}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$128 x^{21} + 7 (64 x^{3 \cdot 6}) {(3 y^{2})}^{1} + 21 {(2 x^{3})}^{5} {(3 y^{2})}^{2} + 35 {(2 x^{3})}^{4} {(3 y^{2})}^{3} + 35 {(2 x^{3})}^{3} {(3 y^{2})}^{4} + 21 {(2 x^{3})}^{2} {(3 y^{2})}^{5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Multiply $3$ by $6$ .

$128 x^{21} + 7 (64 x^{18}) {(3 y^{2})}^{1} + 21 {(2 x^{3})}^{5} {(3 y^{2})}^{2} + 35 {(2 x^{3})}^{4} {(3 y^{2})}^{3} + 35 {(2 x^{3})}^{3} {(3 y^{2})}^{4} + 21 {(2 x^{3})}^{2} {(3 y^{2})}^{5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Multiply $64$ by $7$ .

$128 x^{21} + 448 x^{18} {(3 y^{2})}^{1} + 21 {(2 x^{3})}^{5} {(3 y^{2})}^{2} + 35 {(2 x^{3})}^{4} {(3 y^{2})}^{3} + 35 {(2 x^{3})}^{3} {(3 y^{2})}^{4} + 21 {(2 x^{3})}^{2} {(3 y^{2})}^{5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Simplify.

$128 x^{21} + 448 x^{18} (3 y^{2}) + 21 {(2 x^{3})}^{5} {(3 y^{2})}^{2} + 35 {(2 x^{3})}^{4} {(3 y^{2})}^{3} + 35 {(2 x^{3})}^{3} {(3 y^{2})}^{4} + 21 {(2 x^{3})}^{2} {(3 y^{2})}^{5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Rewrite using the commutative property of multiplication.

$128 x^{21} + 448 \cdot 3 x^{18} y^{2} + 21 {(2 x^{3})}^{5} {(3 y^{2})}^{2} + 35 {(2 x^{3})}^{4} {(3 y^{2})}^{3} + 35 {(2 x^{3})}^{3} {(3 y^{2})}^{4} + 21 {(2 x^{3})}^{2} {(3 y^{2})}^{5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Multiply $448$ by $3$ .

$128 x^{21} + 1344 x^{18} y^{2} + 21 {(2 x^{3})}^{5} {(3 y^{2})}^{2} + 35 {(2 x^{3})}^{4} {(3 y^{2})}^{3} + 35 {(2 x^{3})}^{3} {(3 y^{2})}^{4} + 21 {(2 x^{3})}^{2} {(3 y^{2})}^{5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Apply the product rule to $2 x^{3}$ .

$128 x^{21} + 1344 x^{18} y^{2} + 21 (2^{5} {(x^{3})}^{5}) {(3 y^{2})}^{2} + 35 {(2 x^{3})}^{4} {(3 y^{2})}^{3} + 35 {(2 x^{3})}^{3} {(3 y^{2})}^{4} + 21 {(2 x^{3})}^{2} {(3 y^{2})}^{5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

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

$128 x^{21} + 1344 x^{18} y^{2} + 21 (32 {(x^{3})}^{5}) {(3 y^{2})}^{2} + 35 {(2 x^{3})}^{4} {(3 y^{2})}^{3} + 35 {(2 x^{3})}^{3} {(3 y^{2})}^{4} + 21 {(2 x^{3})}^{2} {(3 y^{2})}^{5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Multiply the exponents in ${(x^{3})}^{5}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$128 x^{21} + 1344 x^{18} y^{2} + 21 (32 x^{3 \cdot 5}) {(3 y^{2})}^{2} + 35 {(2 x^{3})}^{4} {(3 y^{2})}^{3} + 35 {(2 x^{3})}^{3} {(3 y^{2})}^{4} + 21 {(2 x^{3})}^{2} {(3 y^{2})}^{5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Multiply $3$ by $5$ .

$128 x^{21} + 1344 x^{18} y^{2} + 21 (32 x^{15}) {(3 y^{2})}^{2} + 35 {(2 x^{3})}^{4} {(3 y^{2})}^{3} + 35 {(2 x^{3})}^{3} {(3 y^{2})}^{4} + 21 {(2 x^{3})}^{2} {(3 y^{2})}^{5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Multiply $32$ by $21$ .

$128 x^{21} + 1344 x^{18} y^{2} + 672 x^{15} {(3 y^{2})}^{2} + 35 {(2 x^{3})}^{4} {(3 y^{2})}^{3} + 35 {(2 x^{3})}^{3} {(3 y^{2})}^{4} + 21 {(2 x^{3})}^{2} {(3 y^{2})}^{5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Apply the product rule to $3 y^{2}$ .

$128 x^{21} + 1344 x^{18} y^{2} + 672 x^{15} (3^{2} {(y^{2})}^{2}) + 35 {(2 x^{3})}^{4} {(3 y^{2})}^{3} + 35 {(2 x^{3})}^{3} {(3 y^{2})}^{4} + 21 {(2 x^{3})}^{2} {(3 y^{2})}^{5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Rewrite using the commutative property of multiplication.

$128 x^{21} + 1344 x^{18} y^{2} + 672 \cdot 3^{2} x^{15} {(y^{2})}^{2} + 35 {(2 x^{3})}^{4} {(3 y^{2})}^{3} + 35 {(2 x^{3})}^{3} {(3 y^{2})}^{4} + 21 {(2 x^{3})}^{2} {(3 y^{2})}^{5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

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

$128 x^{21} + 1344 x^{18} y^{2} + 672 \cdot 9 x^{15} {(y^{2})}^{2} + 35 {(2 x^{3})}^{4} {(3 y^{2})}^{3} + 35 {(2 x^{3})}^{3} {(3 y^{2})}^{4} + 21 {(2 x^{3})}^{2} {(3 y^{2})}^{5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Multiply $672$ by $9$ .

$128 x^{21} + 1344 x^{18} y^{2} + 6048 x^{15} {(y^{2})}^{2} + 35 {(2 x^{3})}^{4} {(3 y^{2})}^{3} + 35 {(2 x^{3})}^{3} {(3 y^{2})}^{4} + 21 {(2 x^{3})}^{2} {(3 y^{2})}^{5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Multiply the exponents in ${(y^{2})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$128 x^{21} + 1344 x^{18} y^{2} + 6048 x^{15} y^{2 \cdot 2} + 35 {(2 x^{3})}^{4} {(3 y^{2})}^{3} + 35 {(2 x^{3})}^{3} {(3 y^{2})}^{4} + 21 {(2 x^{3})}^{2} {(3 y^{2})}^{5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Multiply $2$ by $2$ .

$128 x^{21} + 1344 x^{18} y^{2} + 6048 x^{15} y^{4} + 35 {(2 x^{3})}^{4} {(3 y^{2})}^{3} + 35 {(2 x^{3})}^{3} {(3 y^{2})}^{4} + 21 {(2 x^{3})}^{2} {(3 y^{2})}^{5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Apply the product rule to $2 x^{3}$ .

$128 x^{21} + 1344 x^{18} y^{2} + 6048 x^{15} y^{4} + 35 (2^{4} {(x^{3})}^{4}) {(3 y^{2})}^{3} + 35 {(2 x^{3})}^{3} {(3 y^{2})}^{4} + 21 {(2 x^{3})}^{2} {(3 y^{2})}^{5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

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

$128 x^{21} + 1344 x^{18} y^{2} + 6048 x^{15} y^{4} + 35 (16 {(x^{3})}^{4}) {(3 y^{2})}^{3} + 35 {(2 x^{3})}^{3} {(3 y^{2})}^{4} + 21 {(2 x^{3})}^{2} {(3 y^{2})}^{5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Multiply the exponents in ${(x^{3})}^{4}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$128 x^{21} + 1344 x^{18} y^{2} + 6048 x^{15} y^{4} + 35 (16 x^{3 \cdot 4}) {(3 y^{2})}^{3} + 35 {(2 x^{3})}^{3} {(3 y^{2})}^{4} + 21 {(2 x^{3})}^{2} {(3 y^{2})}^{5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Multiply $3$ by $4$ .

$128 x^{21} + 1344 x^{18} y^{2} + 6048 x^{15} y^{4} + 35 (16 x^{12}) {(3 y^{2})}^{3} + 35 {(2 x^{3})}^{3} {(3 y^{2})}^{4} + 21 {(2 x^{3})}^{2} {(3 y^{2})}^{5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Multiply $16$ by $35$ .

$128 x^{21} + 1344 x^{18} y^{2} + 6048 x^{15} y^{4} + 560 x^{12} {(3 y^{2})}^{3} + 35 {(2 x^{3})}^{3} {(3 y^{2})}^{4} + 21 {(2 x^{3})}^{2} {(3 y^{2})}^{5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Apply the product rule to $3 y^{2}$ .

$128 x^{21} + 1344 x^{18} y^{2} + 6048 x^{15} y^{4} + 560 x^{12} (3^{3} {(y^{2})}^{3}) + 35 {(2 x^{3})}^{3} {(3 y^{2})}^{4} + 21 {(2 x^{3})}^{2} {(3 y^{2})}^{5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Rewrite using the commutative property of multiplication.

$128 x^{21} + 1344 x^{18} y^{2} + 6048 x^{15} y^{4} + 560 \cdot 3^{3} x^{12} {(y^{2})}^{3} + 35 {(2 x^{3})}^{3} {(3 y^{2})}^{4} + 21 {(2 x^{3})}^{2} {(3 y^{2})}^{5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

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

$128 x^{21} + 1344 x^{18} y^{2} + 6048 x^{15} y^{4} + 560 \cdot 27 x^{12} {(y^{2})}^{3} + 35 {(2 x^{3})}^{3} {(3 y^{2})}^{4} + 21 {(2 x^{3})}^{2} {(3 y^{2})}^{5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Multiply $560$ by $27$ .

$128 x^{21} + 1344 x^{18} y^{2} + 6048 x^{15} y^{4} + 15120 x^{12} {(y^{2})}^{3} + 35 {(2 x^{3})}^{3} {(3 y^{2})}^{4} + 21 {(2 x^{3})}^{2} {(3 y^{2})}^{5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Multiply the exponents in ${(y^{2})}^{3}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$128 x^{21} + 1344 x^{18} y^{2} + 6048 x^{15} y^{4} + 15120 x^{12} y^{2 \cdot 3} + 35 {(2 x^{3})}^{3} {(3 y^{2})}^{4} + 21 {(2 x^{3})}^{2} {(3 y^{2})}^{5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Multiply $2$ by $3$ .

$128 x^{21} + 1344 x^{18} y^{2} + 6048 x^{15} y^{4} + 15120 x^{12} y^{6} + 35 {(2 x^{3})}^{3} {(3 y^{2})}^{4} + 21 {(2 x^{3})}^{2} {(3 y^{2})}^{5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Apply the product rule to $2 x^{3}$ .

$128 x^{21} + 1344 x^{18} y^{2} + 6048 x^{15} y^{4} + 15120 x^{12} y^{6} + 35 (2^{3} {(x^{3})}^{3}) {(3 y^{2})}^{4} + 21 {(2 x^{3})}^{2} {(3 y^{2})}^{5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

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

$128 x^{21} + 1344 x^{18} y^{2} + 6048 x^{15} y^{4} + 15120 x^{12} y^{6} + 35 (8 {(x^{3})}^{3}) {(3 y^{2})}^{4} + 21 {(2 x^{3})}^{2} {(3 y^{2})}^{5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Multiply the exponents in ${(x^{3})}^{3}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$128 x^{21} + 1344 x^{18} y^{2} + 6048 x^{15} y^{4} + 15120 x^{12} y^{6} + 35 (8 x^{3 \cdot 3}) {(3 y^{2})}^{4} + 21 {(2 x^{3})}^{2} {(3 y^{2})}^{5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Multiply $3$ by $3$ .

$128 x^{21} + 1344 x^{18} y^{2} + 6048 x^{15} y^{4} + 15120 x^{12} y^{6} + 35 (8 x^{9}) {(3 y^{2})}^{4} + 21 {(2 x^{3})}^{2} {(3 y^{2})}^{5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Multiply $8$ by $35$ .

$128 x^{21} + 1344 x^{18} y^{2} + 6048 x^{15} y^{4} + 15120 x^{12} y^{6} + 280 x^{9} {(3 y^{2})}^{4} + 21 {(2 x^{3})}^{2} {(3 y^{2})}^{5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Apply the product rule to $3 y^{2}$ .

$128 x^{21} + 1344 x^{18} y^{2} + 6048 x^{15} y^{4} + 15120 x^{12} y^{6} + 280 x^{9} (3^{4} {(y^{2})}^{4}) + 21 {(2 x^{3})}^{2} {(3 y^{2})}^{5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Rewrite using the commutative property of multiplication.

$128 x^{21} + 1344 x^{18} y^{2} + 6048 x^{15} y^{4} + 15120 x^{12} y^{6} + 280 \cdot 3^{4} x^{9} {(y^{2})}^{4} + 21 {(2 x^{3})}^{2} {(3 y^{2})}^{5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

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

$128 x^{21} + 1344 x^{18} y^{2} + 6048 x^{15} y^{4} + 15120 x^{12} y^{6} + 280 \cdot 81 x^{9} {(y^{2})}^{4} + 21 {(2 x^{3})}^{2} {(3 y^{2})}^{5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Multiply $280$ by $81$ .

$128 x^{21} + 1344 x^{18} y^{2} + 6048 x^{15} y^{4} + 15120 x^{12} y^{6} + 22680 x^{9} {(y^{2})}^{4} + 21 {(2 x^{3})}^{2} {(3 y^{2})}^{5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Multiply the exponents in ${(y^{2})}^{4}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$128 x^{21} + 1344 x^{18} y^{2} + 6048 x^{15} y^{4} + 15120 x^{12} y^{6} + 22680 x^{9} y^{2 \cdot 4} + 21 {(2 x^{3})}^{2} {(3 y^{2})}^{5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Multiply $2$ by $4$ .

$128 x^{21} + 1344 x^{18} y^{2} + 6048 x^{15} y^{4} + 15120 x^{12} y^{6} + 22680 x^{9} y^{8} + 21 {(2 x^{3})}^{2} {(3 y^{2})}^{5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Apply the product rule to $2 x^{3}$ .

$128 x^{21} + 1344 x^{18} y^{2} + 6048 x^{15} y^{4} + 15120 x^{12} y^{6} + 22680 x^{9} y^{8} + 21 (2^{2} {(x^{3})}^{2}) {(3 y^{2})}^{5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

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

$128 x^{21} + 1344 x^{18} y^{2} + 6048 x^{15} y^{4} + 15120 x^{12} y^{6} + 22680 x^{9} y^{8} + 21 (4 {(x^{3})}^{2}) {(3 y^{2})}^{5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Multiply the exponents in ${(x^{3})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$128 x^{21} + 1344 x^{18} y^{2} + 6048 x^{15} y^{4} + 15120 x^{12} y^{6} + 22680 x^{9} y^{8} + 21 (4 x^{3 \cdot 2}) {(3 y^{2})}^{5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Multiply $3$ by $2$ .

$128 x^{21} + 1344 x^{18} y^{2} + 6048 x^{15} y^{4} + 15120 x^{12} y^{6} + 22680 x^{9} y^{8} + 21 (4 x^{6}) {(3 y^{2})}^{5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Multiply $4$ by $21$ .

$128 x^{21} + 1344 x^{18} y^{2} + 6048 x^{15} y^{4} + 15120 x^{12} y^{6} + 22680 x^{9} y^{8} + 84 x^{6} {(3 y^{2})}^{5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Apply the product rule to $3 y^{2}$ .

$128 x^{21} + 1344 x^{18} y^{2} + 6048 x^{15} y^{4} + 15120 x^{12} y^{6} + 22680 x^{9} y^{8} + 84 x^{6} (3^{5} {(y^{2})}^{5}) + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Rewrite using the commutative property of multiplication.

$128 x^{21} + 1344 x^{18} y^{2} + 6048 x^{15} y^{4} + 15120 x^{12} y^{6} + 22680 x^{9} y^{8} + 84 \cdot 3^{5} x^{6} {(y^{2})}^{5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

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

$128 x^{21} + 1344 x^{18} y^{2} + 6048 x^{15} y^{4} + 15120 x^{12} y^{6} + 22680 x^{9} y^{8} + 84 \cdot 243 x^{6} {(y^{2})}^{5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Multiply $84$ by $243$ .

$128 x^{21} + 1344 x^{18} y^{2} + 6048 x^{15} y^{4} + 15120 x^{12} y^{6} + 22680 x^{9} y^{8} + 20412 x^{6} {(y^{2})}^{5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Multiply the exponents in ${(y^{2})}^{5}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$128 x^{21} + 1344 x^{18} y^{2} + 6048 x^{15} y^{4} + 15120 x^{12} y^{6} + 22680 x^{9} y^{8} + 20412 x^{6} y^{2 \cdot 5} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Multiply $2$ by $5$ .

$128 x^{21} + 1344 x^{18} y^{2} + 6048 x^{15} y^{4} + 15120 x^{12} y^{6} + 22680 x^{9} y^{8} + 20412 x^{6} y^{10} + 7 {(2 x^{3})}^{1} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Simplify.

$128 x^{21} + 1344 x^{18} y^{2} + 6048 x^{15} y^{4} + 15120 x^{12} y^{6} + 22680 x^{9} y^{8} + 20412 x^{6} y^{10} + 7 (2 x^{3}) {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Multiply $2$ by $7$ .

$128 x^{21} + 1344 x^{18} y^{2} + 6048 x^{15} y^{4} + 15120 x^{12} y^{6} + 22680 x^{9} y^{8} + 20412 x^{6} y^{10} + 14 x^{3} {(3 y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Apply the product rule to $3 y^{2}$ .

$128 x^{21} + 1344 x^{18} y^{2} + 6048 x^{15} y^{4} + 15120 x^{12} y^{6} + 22680 x^{9} y^{8} + 20412 x^{6} y^{10} + 14 x^{3} (3^{6} {(y^{2})}^{6}) + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Rewrite using the commutative property of multiplication.

$128 x^{21} + 1344 x^{18} y^{2} + 6048 x^{15} y^{4} + 15120 x^{12} y^{6} + 22680 x^{9} y^{8} + 20412 x^{6} y^{10} + 14 \cdot 3^{6} x^{3} {(y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

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

$128 x^{21} + 1344 x^{18} y^{2} + 6048 x^{15} y^{4} + 15120 x^{12} y^{6} + 22680 x^{9} y^{8} + 20412 x^{6} y^{10} + 14 \cdot 729 x^{3} {(y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Multiply $14$ by $729$ .

$128 x^{21} + 1344 x^{18} y^{2} + 6048 x^{15} y^{4} + 15120 x^{12} y^{6} + 22680 x^{9} y^{8} + 20412 x^{6} y^{10} + 10206 x^{3} {(y^{2})}^{6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Multiply the exponents in ${(y^{2})}^{6}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$128 x^{21} + 1344 x^{18} y^{2} + 6048 x^{15} y^{4} + 15120 x^{12} y^{6} + 22680 x^{9} y^{8} + 20412 x^{6} y^{10} + 10206 x^{3} y^{2 \cdot 6} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Multiply $2$ by $6$ .

$128 x^{21} + 1344 x^{18} y^{2} + 6048 x^{15} y^{4} + 15120 x^{12} y^{6} + 22680 x^{9} y^{8} + 20412 x^{6} y^{10} + 10206 x^{3} y^{12} + 1 {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Multiply ${(2 x^{3})}^{0}$ by $1$ .

$128 x^{21} + 1344 x^{18} y^{2} + 6048 x^{15} y^{4} + 15120 x^{12} y^{6} + 22680 x^{9} y^{8} + 20412 x^{6} y^{10} + 10206 x^{3} y^{12} + {(2 x^{3})}^{0} {(3 y^{2})}^{7}$

Apply the product rule to $2 x^{3}$ .

$128 x^{21} + 1344 x^{18} y^{2} + 6048 x^{15} y^{4} + 15120 x^{12} y^{6} + 22680 x^{9} y^{8} + 20412 x^{6} y^{10} + 10206 x^{3} y^{12} + 2^{0} {(x^{3})}^{0} {(3 y^{2})}^{7}$

Anything raised to $0$ is $1$ .

$128 x^{21} + 1344 x^{18} y^{2} + 6048 x^{15} y^{4} + 15120 x^{12} y^{6} + 22680 x^{9} y^{8} + 20412 x^{6} y^{10} + 10206 x^{3} y^{12} + 1 {(x^{3})}^{0} {(3 y^{2})}^{7}$

Multiply ${(x^{3})}^{0}$ by $1$ .

$128 x^{21} + 1344 x^{18} y^{2} + 6048 x^{15} y^{4} + 15120 x^{12} y^{6} + 22680 x^{9} y^{8} + 20412 x^{6} y^{10} + 10206 x^{3} y^{12} + {(x^{3})}^{0} {(3 y^{2})}^{7}$

Multiply the exponents in ${(x^{3})}^{0}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$128 x^{21} + 1344 x^{18} y^{2} + 6048 x^{15} y^{4} + 15120 x^{12} y^{6} + 22680 x^{9} y^{8} + 20412 x^{6} y^{10} + 10206 x^{3} y^{12} + x^{3 \cdot 0} {(3 y^{2})}^{7}$

Multiply $3$ by $0$ .

$128 x^{21} + 1344 x^{18} y^{2} + 6048 x^{15} y^{4} + 15120 x^{12} y^{6} + 22680 x^{9} y^{8} + 20412 x^{6} y^{10} + 10206 x^{3} y^{12} + x^{0} {(3 y^{2})}^{7}$

Anything raised to $0$ is $1$ .

$128 x^{21} + 1344 x^{18} y^{2} + 6048 x^{15} y^{4} + 15120 x^{12} y^{6} + 22680 x^{9} y^{8} + 20412 x^{6} y^{10} + 10206 x^{3} y^{12} + 1 {(3 y^{2})}^{7}$

Multiply ${(3 y^{2})}^{7}$ by $1$ .

$128 x^{21} + 1344 x^{18} y^{2} + 6048 x^{15} y^{4} + 15120 x^{12} y^{6} + 22680 x^{9} y^{8} + 20412 x^{6} y^{10} + 10206 x^{3} y^{12} + {(3 y^{2})}^{7}$

Apply the product rule to $3 y^{2}$ .

$128 x^{21} + 1344 x^{18} y^{2} + 6048 x^{15} y^{4} + 15120 x^{12} y^{6} + 22680 x^{9} y^{8} + 20412 x^{6} y^{10} + 10206 x^{3} y^{12} + 3^{7} {(y^{2})}^{7}$

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

$128 x^{21} + 1344 x^{18} y^{2} + 6048 x^{15} y^{4} + 15120 x^{12} y^{6} + 22680 x^{9} y^{8} + 20412 x^{6} y^{10} + 10206 x^{3} y^{12} + 2187 {(y^{2})}^{7}$

Multiply the exponents in ${(y^{2})}^{7}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$128 x^{21} + 1344 x^{18} y^{2} + 6048 x^{15} y^{4} + 15120 x^{12} y^{6} + 22680 x^{9} y^{8} + 20412 x^{6} y^{10} + 10206 x^{3} y^{12} + 2187 y^{2 \cdot 7}$

Multiply $2$ by $7$ .

$128 x^{21} + 1344 x^{18} y^{2} + 6048 x^{15} y^{4} + 15120 x^{12} y^{6} + 22680 x^{9} y^{8} + 20412 x^{6} y^{10} + 10206 x^{3} y^{12} + 2187 y^{14}$

Expand Using Pascal's Triangle (2x^3+3y^2)^7

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