Expand Using Pascal's Triangle (2x+3y)^4

${(2 x + 3 y)}^{4}$

Step 1

Pascal\'s Triangle can be displayed as such:

$1$

$1 - 1$

$1 - 2 - 1$

$1 - 3 - 3 - 1$

$1 - 4 - 6 - 4 - 1$

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 ${(2 x + 3 y)}^{4}$ , $n = 4$ so the coefficients of the expansion will correspond with line $5$ .

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 $1 - 4 - 6 - 4 - 1$ .

$1 a^{4} b^{0} + 4 a^{3} b + 6 a^{2} b^{2} + 4 a b^{3} + 1 a^{0} b^{4}$

Step 3

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

$1 {(2 x)}^{4} {(3 y)}^{0} + 4 {(2 x)}^{3} {(3 y)}^{1} + 6 {(2 x)}^{2} {(3 y)}^{2} + 4 {(2 x)}^{1} {(3 y)}^{3} + 1 {(2 x)}^{0} {(3 y)}^{4}$

Step 4

Simplify each term.

Multiply ${(2 x)}^{4}$ by $1$ .

${(2 x)}^{4} {(3 y)}^{0} + 4 {(2 x)}^{3} {(3 y)}^{1} + 6 {(2 x)}^{2} {(3 y)}^{2} + 4 {(2 x)}^{1} {(3 y)}^{3} + 1 {(2 x)}^{0} {(3 y)}^{4}$

Apply the product rule to $2 x$ .

$2^{4} x^{4} {(3 y)}^{0} + 4 {(2 x)}^{3} {(3 y)}^{1} + 6 {(2 x)}^{2} {(3 y)}^{2} + 4 {(2 x)}^{1} {(3 y)}^{3} + 1 {(2 x)}^{0} {(3 y)}^{4}$

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

$16 x^{4} {(3 y)}^{0} + 4 {(2 x)}^{3} {(3 y)}^{1} + 6 {(2 x)}^{2} {(3 y)}^{2} + 4 {(2 x)}^{1} {(3 y)}^{3} + 1 {(2 x)}^{0} {(3 y)}^{4}$

Apply the product rule to $3 y$ .

$16 x^{4} (3^{0} y^{0}) + 4 {(2 x)}^{3} {(3 y)}^{1} + 6 {(2 x)}^{2} {(3 y)}^{2} + 4 {(2 x)}^{1} {(3 y)}^{3} + 1 {(2 x)}^{0} {(3 y)}^{4}$

Rewrite using the commutative property of multiplication.

$16 \cdot 3^{0} x^{4} y^{0} + 4 {(2 x)}^{3} {(3 y)}^{1} + 6 {(2 x)}^{2} {(3 y)}^{2} + 4 {(2 x)}^{1} {(3 y)}^{3} + 1 {(2 x)}^{0} {(3 y)}^{4}$

Anything raised to $0$ is $1$ .

$16 \cdot 1 x^{4} y^{0} + 4 {(2 x)}^{3} {(3 y)}^{1} + 6 {(2 x)}^{2} {(3 y)}^{2} + 4 {(2 x)}^{1} {(3 y)}^{3} + 1 {(2 x)}^{0} {(3 y)}^{4}$

Multiply $16$ by $1$ .

$16 x^{4} y^{0} + 4 {(2 x)}^{3} {(3 y)}^{1} + 6 {(2 x)}^{2} {(3 y)}^{2} + 4 {(2 x)}^{1} {(3 y)}^{3} + 1 {(2 x)}^{0} {(3 y)}^{4}$

Anything raised to $0$ is $1$ .

$16 x^{4} \cdot 1 + 4 {(2 x)}^{3} {(3 y)}^{1} + 6 {(2 x)}^{2} {(3 y)}^{2} + 4 {(2 x)}^{1} {(3 y)}^{3} + 1 {(2 x)}^{0} {(3 y)}^{4}$

Multiply $16$ by $1$ .

$16 x^{4} + 4 {(2 x)}^{3} {(3 y)}^{1} + 6 {(2 x)}^{2} {(3 y)}^{2} + 4 {(2 x)}^{1} {(3 y)}^{3} + 1 {(2 x)}^{0} {(3 y)}^{4}$

Apply the product rule to $2 x$ .

$16 x^{4} + 4 (2^{3} x^{3}) {(3 y)}^{1} + 6 {(2 x)}^{2} {(3 y)}^{2} + 4 {(2 x)}^{1} {(3 y)}^{3} + 1 {(2 x)}^{0} {(3 y)}^{4}$

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

$16 x^{4} + 4 (8 x^{3}) {(3 y)}^{1} + 6 {(2 x)}^{2} {(3 y)}^{2} + 4 {(2 x)}^{1} {(3 y)}^{3} + 1 {(2 x)}^{0} {(3 y)}^{4}$

Multiply $8$ by $4$ .

$16 x^{4} + 32 x^{3} {(3 y)}^{1} + 6 {(2 x)}^{2} {(3 y)}^{2} + 4 {(2 x)}^{1} {(3 y)}^{3} + 1 {(2 x)}^{0} {(3 y)}^{4}$

Simplify.

$16 x^{4} + 32 x^{3} (3 y) + 6 {(2 x)}^{2} {(3 y)}^{2} + 4 {(2 x)}^{1} {(3 y)}^{3} + 1 {(2 x)}^{0} {(3 y)}^{4}$

Rewrite using the commutative property of multiplication.

$16 x^{4} + 32 \cdot 3 x^{3} y + 6 {(2 x)}^{2} {(3 y)}^{2} + 4 {(2 x)}^{1} {(3 y)}^{3} + 1 {(2 x)}^{0} {(3 y)}^{4}$

Multiply $32$ by $3$ .

$16 x^{4} + 96 x^{3} y + 6 {(2 x)}^{2} {(3 y)}^{2} + 4 {(2 x)}^{1} {(3 y)}^{3} + 1 {(2 x)}^{0} {(3 y)}^{4}$

Apply the product rule to $2 x$ .

$16 x^{4} + 96 x^{3} y + 6 (2^{2} x^{2}) {(3 y)}^{2} + 4 {(2 x)}^{1} {(3 y)}^{3} + 1 {(2 x)}^{0} {(3 y)}^{4}$

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

$16 x^{4} + 96 x^{3} y + 6 (4 x^{2}) {(3 y)}^{2} + 4 {(2 x)}^{1} {(3 y)}^{3} + 1 {(2 x)}^{0} {(3 y)}^{4}$

Multiply $4$ by $6$ .

$16 x^{4} + 96 x^{3} y + 24 x^{2} {(3 y)}^{2} + 4 {(2 x)}^{1} {(3 y)}^{3} + 1 {(2 x)}^{0} {(3 y)}^{4}$

Apply the product rule to $3 y$ .

$16 x^{4} + 96 x^{3} y + 24 x^{2} (3^{2} y^{2}) + 4 {(2 x)}^{1} {(3 y)}^{3} + 1 {(2 x)}^{0} {(3 y)}^{4}$

Rewrite using the commutative property of multiplication.

$16 x^{4} + 96 x^{3} y + 24 \cdot 3^{2} x^{2} y^{2} + 4 {(2 x)}^{1} {(3 y)}^{3} + 1 {(2 x)}^{0} {(3 y)}^{4}$

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

$16 x^{4} + 96 x^{3} y + 24 \cdot 9 x^{2} y^{2} + 4 {(2 x)}^{1} {(3 y)}^{3} + 1 {(2 x)}^{0} {(3 y)}^{4}$

Multiply $24$ by $9$ .

$16 x^{4} + 96 x^{3} y + 216 x^{2} y^{2} + 4 {(2 x)}^{1} {(3 y)}^{3} + 1 {(2 x)}^{0} {(3 y)}^{4}$

Simplify.

$16 x^{4} + 96 x^{3} y + 216 x^{2} y^{2} + 4 (2 x) {(3 y)}^{3} + 1 {(2 x)}^{0} {(3 y)}^{4}$

Multiply $2$ by $4$ .

$16 x^{4} + 96 x^{3} y + 216 x^{2} y^{2} + 8 x {(3 y)}^{3} + 1 {(2 x)}^{0} {(3 y)}^{4}$

Apply the product rule to $3 y$ .

$16 x^{4} + 96 x^{3} y + 216 x^{2} y^{2} + 8 x (3^{3} y^{3}) + 1 {(2 x)}^{0} {(3 y)}^{4}$

Rewrite using the commutative property of multiplication.

$16 x^{4} + 96 x^{3} y + 216 x^{2} y^{2} + 8 \cdot 3^{3} x y^{3} + 1 {(2 x)}^{0} {(3 y)}^{4}$

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

$16 x^{4} + 96 x^{3} y + 216 x^{2} y^{2} + 8 \cdot 27 x y^{3} + 1 {(2 x)}^{0} {(3 y)}^{4}$

Multiply $8$ by $27$ .

$16 x^{4} + 96 x^{3} y + 216 x^{2} y^{2} + 216 x y^{3} + 1 {(2 x)}^{0} {(3 y)}^{4}$

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

$16 x^{4} + 96 x^{3} y + 216 x^{2} y^{2} + 216 x y^{3} + {(2 x)}^{0} {(3 y)}^{4}$

Apply the product rule to $2 x$ .

$16 x^{4} + 96 x^{3} y + 216 x^{2} y^{2} + 216 x y^{3} + 2^{0} x^{0} {(3 y)}^{4}$

Anything raised to $0$ is $1$ .

$16 x^{4} + 96 x^{3} y + 216 x^{2} y^{2} + 216 x y^{3} + 1 x^{0} {(3 y)}^{4}$

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

$16 x^{4} + 96 x^{3} y + 216 x^{2} y^{2} + 216 x y^{3} + x^{0} {(3 y)}^{4}$

Anything raised to $0$ is $1$ .

$16 x^{4} + 96 x^{3} y + 216 x^{2} y^{2} + 216 x y^{3} + 1 {(3 y)}^{4}$

Multiply ${(3 y)}^{4}$ by $1$ .

$16 x^{4} + 96 x^{3} y + 216 x^{2} y^{2} + 216 x y^{3} + {(3 y)}^{4}$

Apply the product rule to $3 y$ .

$16 x^{4} + 96 x^{3} y + 216 x^{2} y^{2} + 216 x y^{3} + 3^{4} y^{4}$

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

$16 x^{4} + 96 x^{3} y + 216 x^{2} y^{2} + 216 x y^{3} + 81 y^{4}$

Expand Using Pascal's Triangle (2x+3y)^4

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