Expand Using Pascal's Triangle (x+2y)^9

${(x + 2 y)}^{9}$

Step 1

Pascal\'s Triangle can be displayed as such:

$1$

$1 - 1$

$1 - 2 - 1$

$\dots$

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

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

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

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 ${(x + 2 y)}^{9}$ , $n = 9$ so the coefficients of the expansion will correspond with line $10$ .

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

$1 a^{9} b^{0} + 9 a^{8} b + 36 a^{7} b^{2} + 84 a^{6} b^{3} + 126 a^{5} b^{4} + 126 a^{4} b^{5} + 84 a^{3} b^{6} + 36 a^{2} b^{7} + 9 a b^{8} + 1 a^{0} b^{9}$

Step 3

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

$1 {(x)}^{9} {(2 y)}^{0} + 9 {(x)}^{8} {(2 y)}^{1} + 36 {(x)}^{7} {(2 y)}^{2} + 84 {(x)}^{6} {(2 y)}^{3} + 126 {(x)}^{5} {(2 y)}^{4} + 126 {(x)}^{4} {(2 y)}^{5} + 84 {(x)}^{3} {(2 y)}^{6} + 36 {(x)}^{2} {(2 y)}^{7} + 9 {(x)}^{1} {(2 y)}^{8} + 1 {(x)}^{0} {(2 y)}^{9}$

Step 4

Simplify each term.

Multiply ${(x)}^{9}$ by $1$ .

${(x)}^{9} {(2 y)}^{0} + 9 {(x)}^{8} {(2 y)}^{1} + 36 {(x)}^{7} {(2 y)}^{2} + 84 {(x)}^{6} {(2 y)}^{3} + 126 {(x)}^{5} {(2 y)}^{4} + 126 {(x)}^{4} {(2 y)}^{5} + 84 {(x)}^{3} {(2 y)}^{6} + 36 {(x)}^{2} {(2 y)}^{7} + 9 {(x)}^{1} {(2 y)}^{8} + 1 {(x)}^{0} {(2 y)}^{9}$

Apply the product rule to $2 y$ .

$x^{9} (2^{0} y^{0}) + 9 {(x)}^{8} {(2 y)}^{1} + 36 {(x)}^{7} {(2 y)}^{2} + 84 {(x)}^{6} {(2 y)}^{3} + 126 {(x)}^{5} {(2 y)}^{4} + 126 {(x)}^{4} {(2 y)}^{5} + 84 {(x)}^{3} {(2 y)}^{6} + 36 {(x)}^{2} {(2 y)}^{7} + 9 {(x)}^{1} {(2 y)}^{8} + 1 {(x)}^{0} {(2 y)}^{9}$

Rewrite using the commutative property of multiplication.

$2^{0} x^{9} y^{0} + 9 {(x)}^{8} {(2 y)}^{1} + 36 {(x)}^{7} {(2 y)}^{2} + 84 {(x)}^{6} {(2 y)}^{3} + 126 {(x)}^{5} {(2 y)}^{4} + 126 {(x)}^{4} {(2 y)}^{5} + 84 {(x)}^{3} {(2 y)}^{6} + 36 {(x)}^{2} {(2 y)}^{7} + 9 {(x)}^{1} {(2 y)}^{8} + 1 {(x)}^{0} {(2 y)}^{9}$

Anything raised to $0$ is $1$ .

$1 x^{9} y^{0} + 9 {(x)}^{8} {(2 y)}^{1} + 36 {(x)}^{7} {(2 y)}^{2} + 84 {(x)}^{6} {(2 y)}^{3} + 126 {(x)}^{5} {(2 y)}^{4} + 126 {(x)}^{4} {(2 y)}^{5} + 84 {(x)}^{3} {(2 y)}^{6} + 36 {(x)}^{2} {(2 y)}^{7} + 9 {(x)}^{1} {(2 y)}^{8} + 1 {(x)}^{0} {(2 y)}^{9}$

Multiply $x^{9}$ by $1$ .

$x^{9} y^{0} + 9 {(x)}^{8} {(2 y)}^{1} + 36 {(x)}^{7} {(2 y)}^{2} + 84 {(x)}^{6} {(2 y)}^{3} + 126 {(x)}^{5} {(2 y)}^{4} + 126 {(x)}^{4} {(2 y)}^{5} + 84 {(x)}^{3} {(2 y)}^{6} + 36 {(x)}^{2} {(2 y)}^{7} + 9 {(x)}^{1} {(2 y)}^{8} + 1 {(x)}^{0} {(2 y)}^{9}$

Anything raised to $0$ is $1$ .

$x^{9} \cdot 1 + 9 {(x)}^{8} {(2 y)}^{1} + 36 {(x)}^{7} {(2 y)}^{2} + 84 {(x)}^{6} {(2 y)}^{3} + 126 {(x)}^{5} {(2 y)}^{4} + 126 {(x)}^{4} {(2 y)}^{5} + 84 {(x)}^{3} {(2 y)}^{6} + 36 {(x)}^{2} {(2 y)}^{7} + 9 {(x)}^{1} {(2 y)}^{8} + 1 {(x)}^{0} {(2 y)}^{9}$

Multiply $x^{9}$ by $1$ .

$x^{9} + 9 {(x)}^{8} {(2 y)}^{1} + 36 {(x)}^{7} {(2 y)}^{2} + 84 {(x)}^{6} {(2 y)}^{3} + 126 {(x)}^{5} {(2 y)}^{4} + 126 {(x)}^{4} {(2 y)}^{5} + 84 {(x)}^{3} {(2 y)}^{6} + 36 {(x)}^{2} {(2 y)}^{7} + 9 {(x)}^{1} {(2 y)}^{8} + 1 {(x)}^{0} {(2 y)}^{9}$

Simplify.

$x^{9} + 9 x^{8} (2 y) + 36 {(x)}^{7} {(2 y)}^{2} + 84 {(x)}^{6} {(2 y)}^{3} + 126 {(x)}^{5} {(2 y)}^{4} + 126 {(x)}^{4} {(2 y)}^{5} + 84 {(x)}^{3} {(2 y)}^{6} + 36 {(x)}^{2} {(2 y)}^{7} + 9 {(x)}^{1} {(2 y)}^{8} + 1 {(x)}^{0} {(2 y)}^{9}$

Rewrite using the commutative property of multiplication.

$x^{9} + 9 \cdot 2 x^{8} y + 36 {(x)}^{7} {(2 y)}^{2} + 84 {(x)}^{6} {(2 y)}^{3} + 126 {(x)}^{5} {(2 y)}^{4} + 126 {(x)}^{4} {(2 y)}^{5} + 84 {(x)}^{3} {(2 y)}^{6} + 36 {(x)}^{2} {(2 y)}^{7} + 9 {(x)}^{1} {(2 y)}^{8} + 1 {(x)}^{0} {(2 y)}^{9}$

Multiply $9$ by $2$ .

$x^{9} + 18 x^{8} y + 36 {(x)}^{7} {(2 y)}^{2} + 84 {(x)}^{6} {(2 y)}^{3} + 126 {(x)}^{5} {(2 y)}^{4} + 126 {(x)}^{4} {(2 y)}^{5} + 84 {(x)}^{3} {(2 y)}^{6} + 36 {(x)}^{2} {(2 y)}^{7} + 9 {(x)}^{1} {(2 y)}^{8} + 1 {(x)}^{0} {(2 y)}^{9}$

Apply the product rule to $2 y$ .

$x^{9} + 18 x^{8} y + 36 x^{7} (2^{2} y^{2}) + 84 {(x)}^{6} {(2 y)}^{3} + 126 {(x)}^{5} {(2 y)}^{4} + 126 {(x)}^{4} {(2 y)}^{5} + 84 {(x)}^{3} {(2 y)}^{6} + 36 {(x)}^{2} {(2 y)}^{7} + 9 {(x)}^{1} {(2 y)}^{8} + 1 {(x)}^{0} {(2 y)}^{9}$

Rewrite using the commutative property of multiplication.

$x^{9} + 18 x^{8} y + 36 \cdot 2^{2} x^{7} y^{2} + 84 {(x)}^{6} {(2 y)}^{3} + 126 {(x)}^{5} {(2 y)}^{4} + 126 {(x)}^{4} {(2 y)}^{5} + 84 {(x)}^{3} {(2 y)}^{6} + 36 {(x)}^{2} {(2 y)}^{7} + 9 {(x)}^{1} {(2 y)}^{8} + 1 {(x)}^{0} {(2 y)}^{9}$

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

$x^{9} + 18 x^{8} y + 36 \cdot 4 x^{7} y^{2} + 84 {(x)}^{6} {(2 y)}^{3} + 126 {(x)}^{5} {(2 y)}^{4} + 126 {(x)}^{4} {(2 y)}^{5} + 84 {(x)}^{3} {(2 y)}^{6} + 36 {(x)}^{2} {(2 y)}^{7} + 9 {(x)}^{1} {(2 y)}^{8} + 1 {(x)}^{0} {(2 y)}^{9}$

Multiply $36$ by $4$ .

$x^{9} + 18 x^{8} y + 144 x^{7} y^{2} + 84 {(x)}^{6} {(2 y)}^{3} + 126 {(x)}^{5} {(2 y)}^{4} + 126 {(x)}^{4} {(2 y)}^{5} + 84 {(x)}^{3} {(2 y)}^{6} + 36 {(x)}^{2} {(2 y)}^{7} + 9 {(x)}^{1} {(2 y)}^{8} + 1 {(x)}^{0} {(2 y)}^{9}$

Apply the product rule to $2 y$ .

$x^{9} + 18 x^{8} y + 144 x^{7} y^{2} + 84 x^{6} (2^{3} y^{3}) + 126 {(x)}^{5} {(2 y)}^{4} + 126 {(x)}^{4} {(2 y)}^{5} + 84 {(x)}^{3} {(2 y)}^{6} + 36 {(x)}^{2} {(2 y)}^{7} + 9 {(x)}^{1} {(2 y)}^{8} + 1 {(x)}^{0} {(2 y)}^{9}$

Rewrite using the commutative property of multiplication.

$x^{9} + 18 x^{8} y + 144 x^{7} y^{2} + 84 \cdot 2^{3} x^{6} y^{3} + 126 {(x)}^{5} {(2 y)}^{4} + 126 {(x)}^{4} {(2 y)}^{5} + 84 {(x)}^{3} {(2 y)}^{6} + 36 {(x)}^{2} {(2 y)}^{7} + 9 {(x)}^{1} {(2 y)}^{8} + 1 {(x)}^{0} {(2 y)}^{9}$

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

$x^{9} + 18 x^{8} y + 144 x^{7} y^{2} + 84 \cdot 8 x^{6} y^{3} + 126 {(x)}^{5} {(2 y)}^{4} + 126 {(x)}^{4} {(2 y)}^{5} + 84 {(x)}^{3} {(2 y)}^{6} + 36 {(x)}^{2} {(2 y)}^{7} + 9 {(x)}^{1} {(2 y)}^{8} + 1 {(x)}^{0} {(2 y)}^{9}$

Multiply $84$ by $8$ .

$x^{9} + 18 x^{8} y + 144 x^{7} y^{2} + 672 x^{6} y^{3} + 126 {(x)}^{5} {(2 y)}^{4} + 126 {(x)}^{4} {(2 y)}^{5} + 84 {(x)}^{3} {(2 y)}^{6} + 36 {(x)}^{2} {(2 y)}^{7} + 9 {(x)}^{1} {(2 y)}^{8} + 1 {(x)}^{0} {(2 y)}^{9}$

Apply the product rule to $2 y$ .

$x^{9} + 18 x^{8} y + 144 x^{7} y^{2} + 672 x^{6} y^{3} + 126 x^{5} (2^{4} y^{4}) + 126 {(x)}^{4} {(2 y)}^{5} + 84 {(x)}^{3} {(2 y)}^{6} + 36 {(x)}^{2} {(2 y)}^{7} + 9 {(x)}^{1} {(2 y)}^{8} + 1 {(x)}^{0} {(2 y)}^{9}$

Rewrite using the commutative property of multiplication.

$x^{9} + 18 x^{8} y + 144 x^{7} y^{2} + 672 x^{6} y^{3} + 126 \cdot 2^{4} x^{5} y^{4} + 126 {(x)}^{4} {(2 y)}^{5} + 84 {(x)}^{3} {(2 y)}^{6} + 36 {(x)}^{2} {(2 y)}^{7} + 9 {(x)}^{1} {(2 y)}^{8} + 1 {(x)}^{0} {(2 y)}^{9}$

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

$x^{9} + 18 x^{8} y + 144 x^{7} y^{2} + 672 x^{6} y^{3} + 126 \cdot 16 x^{5} y^{4} + 126 {(x)}^{4} {(2 y)}^{5} + 84 {(x)}^{3} {(2 y)}^{6} + 36 {(x)}^{2} {(2 y)}^{7} + 9 {(x)}^{1} {(2 y)}^{8} + 1 {(x)}^{0} {(2 y)}^{9}$

Multiply $126$ by $16$ .

$x^{9} + 18 x^{8} y + 144 x^{7} y^{2} + 672 x^{6} y^{3} + 2016 x^{5} y^{4} + 126 {(x)}^{4} {(2 y)}^{5} + 84 {(x)}^{3} {(2 y)}^{6} + 36 {(x)}^{2} {(2 y)}^{7} + 9 {(x)}^{1} {(2 y)}^{8} + 1 {(x)}^{0} {(2 y)}^{9}$

Apply the product rule to $2 y$ .

$x^{9} + 18 x^{8} y + 144 x^{7} y^{2} + 672 x^{6} y^{3} + 2016 x^{5} y^{4} + 126 x^{4} (2^{5} y^{5}) + 84 {(x)}^{3} {(2 y)}^{6} + 36 {(x)}^{2} {(2 y)}^{7} + 9 {(x)}^{1} {(2 y)}^{8} + 1 {(x)}^{0} {(2 y)}^{9}$

Rewrite using the commutative property of multiplication.

$x^{9} + 18 x^{8} y + 144 x^{7} y^{2} + 672 x^{6} y^{3} + 2016 x^{5} y^{4} + 126 \cdot 2^{5} x^{4} y^{5} + 84 {(x)}^{3} {(2 y)}^{6} + 36 {(x)}^{2} {(2 y)}^{7} + 9 {(x)}^{1} {(2 y)}^{8} + 1 {(x)}^{0} {(2 y)}^{9}$

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

$x^{9} + 18 x^{8} y + 144 x^{7} y^{2} + 672 x^{6} y^{3} + 2016 x^{5} y^{4} + 126 \cdot 32 x^{4} y^{5} + 84 {(x)}^{3} {(2 y)}^{6} + 36 {(x)}^{2} {(2 y)}^{7} + 9 {(x)}^{1} {(2 y)}^{8} + 1 {(x)}^{0} {(2 y)}^{9}$

Multiply $126$ by $32$ .

$x^{9} + 18 x^{8} y + 144 x^{7} y^{2} + 672 x^{6} y^{3} + 2016 x^{5} y^{4} + 4032 x^{4} y^{5} + 84 {(x)}^{3} {(2 y)}^{6} + 36 {(x)}^{2} {(2 y)}^{7} + 9 {(x)}^{1} {(2 y)}^{8} + 1 {(x)}^{0} {(2 y)}^{9}$

Apply the product rule to $2 y$ .

$x^{9} + 18 x^{8} y + 144 x^{7} y^{2} + 672 x^{6} y^{3} + 2016 x^{5} y^{4} + 4032 x^{4} y^{5} + 84 x^{3} (2^{6} y^{6}) + 36 {(x)}^{2} {(2 y)}^{7} + 9 {(x)}^{1} {(2 y)}^{8} + 1 {(x)}^{0} {(2 y)}^{9}$

Rewrite using the commutative property of multiplication.

$x^{9} + 18 x^{8} y + 144 x^{7} y^{2} + 672 x^{6} y^{3} + 2016 x^{5} y^{4} + 4032 x^{4} y^{5} + 84 \cdot 2^{6} x^{3} y^{6} + 36 {(x)}^{2} {(2 y)}^{7} + 9 {(x)}^{1} {(2 y)}^{8} + 1 {(x)}^{0} {(2 y)}^{9}$

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

$x^{9} + 18 x^{8} y + 144 x^{7} y^{2} + 672 x^{6} y^{3} + 2016 x^{5} y^{4} + 4032 x^{4} y^{5} + 84 \cdot 64 x^{3} y^{6} + 36 {(x)}^{2} {(2 y)}^{7} + 9 {(x)}^{1} {(2 y)}^{8} + 1 {(x)}^{0} {(2 y)}^{9}$

Multiply $84$ by $64$ .

$x^{9} + 18 x^{8} y + 144 x^{7} y^{2} + 672 x^{6} y^{3} + 2016 x^{5} y^{4} + 4032 x^{4} y^{5} + 5376 x^{3} y^{6} + 36 {(x)}^{2} {(2 y)}^{7} + 9 {(x)}^{1} {(2 y)}^{8} + 1 {(x)}^{0} {(2 y)}^{9}$

Apply the product rule to $2 y$ .

$x^{9} + 18 x^{8} y + 144 x^{7} y^{2} + 672 x^{6} y^{3} + 2016 x^{5} y^{4} + 4032 x^{4} y^{5} + 5376 x^{3} y^{6} + 36 x^{2} (2^{7} y^{7}) + 9 {(x)}^{1} {(2 y)}^{8} + 1 {(x)}^{0} {(2 y)}^{9}$

Rewrite using the commutative property of multiplication.

$x^{9} + 18 x^{8} y + 144 x^{7} y^{2} + 672 x^{6} y^{3} + 2016 x^{5} y^{4} + 4032 x^{4} y^{5} + 5376 x^{3} y^{6} + 36 \cdot 2^{7} x^{2} y^{7} + 9 {(x)}^{1} {(2 y)}^{8} + 1 {(x)}^{0} {(2 y)}^{9}$

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

$x^{9} + 18 x^{8} y + 144 x^{7} y^{2} + 672 x^{6} y^{3} + 2016 x^{5} y^{4} + 4032 x^{4} y^{5} + 5376 x^{3} y^{6} + 36 \cdot 128 x^{2} y^{7} + 9 {(x)}^{1} {(2 y)}^{8} + 1 {(x)}^{0} {(2 y)}^{9}$

Multiply $36$ by $128$ .

$x^{9} + 18 x^{8} y + 144 x^{7} y^{2} + 672 x^{6} y^{3} + 2016 x^{5} y^{4} + 4032 x^{4} y^{5} + 5376 x^{3} y^{6} + 4608 x^{2} y^{7} + 9 {(x)}^{1} {(2 y)}^{8} + 1 {(x)}^{0} {(2 y)}^{9}$

Simplify.

$x^{9} + 18 x^{8} y + 144 x^{7} y^{2} + 672 x^{6} y^{3} + 2016 x^{5} y^{4} + 4032 x^{4} y^{5} + 5376 x^{3} y^{6} + 4608 x^{2} y^{7} + 9 x {(2 y)}^{8} + 1 {(x)}^{0} {(2 y)}^{9}$

Apply the product rule to $2 y$ .

$x^{9} + 18 x^{8} y + 144 x^{7} y^{2} + 672 x^{6} y^{3} + 2016 x^{5} y^{4} + 4032 x^{4} y^{5} + 5376 x^{3} y^{6} + 4608 x^{2} y^{7} + 9 x (2^{8} y^{8}) + 1 {(x)}^{0} {(2 y)}^{9}$

Rewrite using the commutative property of multiplication.

$x^{9} + 18 x^{8} y + 144 x^{7} y^{2} + 672 x^{6} y^{3} + 2016 x^{5} y^{4} + 4032 x^{4} y^{5} + 5376 x^{3} y^{6} + 4608 x^{2} y^{7} + 9 \cdot 2^{8} x y^{8} + 1 {(x)}^{0} {(2 y)}^{9}$

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

$x^{9} + 18 x^{8} y + 144 x^{7} y^{2} + 672 x^{6} y^{3} + 2016 x^{5} y^{4} + 4032 x^{4} y^{5} + 5376 x^{3} y^{6} + 4608 x^{2} y^{7} + 9 \cdot 256 x y^{8} + 1 {(x)}^{0} {(2 y)}^{9}$

Multiply $9$ by $256$ .

$x^{9} + 18 x^{8} y + 144 x^{7} y^{2} + 672 x^{6} y^{3} + 2016 x^{5} y^{4} + 4032 x^{4} y^{5} + 5376 x^{3} y^{6} + 4608 x^{2} y^{7} + 2304 x y^{8} + 1 {(x)}^{0} {(2 y)}^{9}$

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

$x^{9} + 18 x^{8} y + 144 x^{7} y^{2} + 672 x^{6} y^{3} + 2016 x^{5} y^{4} + 4032 x^{4} y^{5} + 5376 x^{3} y^{6} + 4608 x^{2} y^{7} + 2304 x y^{8} + {(x)}^{0} {(2 y)}^{9}$

Anything raised to $0$ is $1$ .

$x^{9} + 18 x^{8} y + 144 x^{7} y^{2} + 672 x^{6} y^{3} + 2016 x^{5} y^{4} + 4032 x^{4} y^{5} + 5376 x^{3} y^{6} + 4608 x^{2} y^{7} + 2304 x y^{8} + 1 {(2 y)}^{9}$

Multiply ${(2 y)}^{9}$ by $1$ .

$x^{9} + 18 x^{8} y + 144 x^{7} y^{2} + 672 x^{6} y^{3} + 2016 x^{5} y^{4} + 4032 x^{4} y^{5} + 5376 x^{3} y^{6} + 4608 x^{2} y^{7} + 2304 x y^{8} + {(2 y)}^{9}$

Apply the product rule to $2 y$ .

$x^{9} + 18 x^{8} y + 144 x^{7} y^{2} + 672 x^{6} y^{3} + 2016 x^{5} y^{4} + 4032 x^{4} y^{5} + 5376 x^{3} y^{6} + 4608 x^{2} y^{7} + 2304 x y^{8} + 2^{9} y^{9}$

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

$x^{9} + 18 x^{8} y + 144 x^{7} y^{2} + 672 x^{6} y^{3} + 2016 x^{5} y^{4} + 4032 x^{4} y^{5} + 5376 x^{3} y^{6} + 4608 x^{2} y^{7} + 2304 x y^{8} + 512 y^{9}$

Expand Using Pascal's Triangle (x+2y)^9

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