Precalculus Examples

${(x + 2 y)}^{10}$

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 ${(x + 2 y)}^{10}$ , $n = 10$ 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 $a$ $x$ and $b$ $2 y$ into the expression.

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

Step 4

Simplify each term.

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Step 4.1

Multiply ${(x)}^{10}$ by $1$ .

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

Step 4.2

Apply the product rule to $2 y$ .

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

Step 4.3

Rewrite using the commutative property of multiplication.

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

Step 4.4

Anything raised to $0$ is $1$ .

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

Step 4.5

Multiply $x^{10}$ by $1$ .

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

Step 4.6

Anything raised to $0$ is $1$ .

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

Step 4.7

Multiply $x^{10}$ by $1$ .

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

Step 4.8

Simplify.

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

Step 4.9

Rewrite using the commutative property of multiplication.

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

Step 4.10

Multiply $10$ by $2$ .

$x^{10} + 20 x^{9} y + 45 {(x)}^{8} {(2 y)}^{2} + 120 {(x)}^{7} {(2 y)}^{3} + 210 {(x)}^{6} {(2 y)}^{4} + 252 {(x)}^{5} {(2 y)}^{5} + 210 {(x)}^{4} {(2 y)}^{6} + 120 {(x)}^{3} {(2 y)}^{7} + 45 {(x)}^{2} {(2 y)}^{8} + 10 {(x)}^{1} {(2 y)}^{9} + 1 {(x)}^{0} {(2 y)}^{10}$

Step 4.11

Apply the product rule to $2 y$ .

$x^{10} + 20 x^{9} y + 45 x^{8} (2^{2} y^{2}) + 120 {(x)}^{7} {(2 y)}^{3} + 210 {(x)}^{6} {(2 y)}^{4} + 252 {(x)}^{5} {(2 y)}^{5} + 210 {(x)}^{4} {(2 y)}^{6} + 120 {(x)}^{3} {(2 y)}^{7} + 45 {(x)}^{2} {(2 y)}^{8} + 10 {(x)}^{1} {(2 y)}^{9} + 1 {(x)}^{0} {(2 y)}^{10}$

Step 4.12

Rewrite using the commutative property of multiplication.

$x^{10} + 20 x^{9} y + 45 \cdot 2^{2} x^{8} y^{2} + 120 {(x)}^{7} {(2 y)}^{3} + 210 {(x)}^{6} {(2 y)}^{4} + 252 {(x)}^{5} {(2 y)}^{5} + 210 {(x)}^{4} {(2 y)}^{6} + 120 {(x)}^{3} {(2 y)}^{7} + 45 {(x)}^{2} {(2 y)}^{8} + 10 {(x)}^{1} {(2 y)}^{9} + 1 {(x)}^{0} {(2 y)}^{10}$

Step 4.13

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

$x^{10} + 20 x^{9} y + 45 \cdot 4 x^{8} y^{2} + 120 {(x)}^{7} {(2 y)}^{3} + 210 {(x)}^{6} {(2 y)}^{4} + 252 {(x)}^{5} {(2 y)}^{5} + 210 {(x)}^{4} {(2 y)}^{6} + 120 {(x)}^{3} {(2 y)}^{7} + 45 {(x)}^{2} {(2 y)}^{8} + 10 {(x)}^{1} {(2 y)}^{9} + 1 {(x)}^{0} {(2 y)}^{10}$

Step 4.14

Multiply $45$ by $4$ .

$x^{10} + 20 x^{9} y + 180 x^{8} y^{2} + 120 {(x)}^{7} {(2 y)}^{3} + 210 {(x)}^{6} {(2 y)}^{4} + 252 {(x)}^{5} {(2 y)}^{5} + 210 {(x)}^{4} {(2 y)}^{6} + 120 {(x)}^{3} {(2 y)}^{7} + 45 {(x)}^{2} {(2 y)}^{8} + 10 {(x)}^{1} {(2 y)}^{9} + 1 {(x)}^{0} {(2 y)}^{10}$

Step 4.15

Apply the product rule to $2 y$ .

$x^{10} + 20 x^{9} y + 180 x^{8} y^{2} + 120 x^{7} (2^{3} y^{3}) + 210 {(x)}^{6} {(2 y)}^{4} + 252 {(x)}^{5} {(2 y)}^{5} + 210 {(x)}^{4} {(2 y)}^{6} + 120 {(x)}^{3} {(2 y)}^{7} + 45 {(x)}^{2} {(2 y)}^{8} + 10 {(x)}^{1} {(2 y)}^{9} + 1 {(x)}^{0} {(2 y)}^{10}$

Step 4.16

Rewrite using the commutative property of multiplication.

$x^{10} + 20 x^{9} y + 180 x^{8} y^{2} + 120 \cdot 2^{3} x^{7} y^{3} + 210 {(x)}^{6} {(2 y)}^{4} + 252 {(x)}^{5} {(2 y)}^{5} + 210 {(x)}^{4} {(2 y)}^{6} + 120 {(x)}^{3} {(2 y)}^{7} + 45 {(x)}^{2} {(2 y)}^{8} + 10 {(x)}^{1} {(2 y)}^{9} + 1 {(x)}^{0} {(2 y)}^{10}$

Step 4.17

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

$x^{10} + 20 x^{9} y + 180 x^{8} y^{2} + 120 \cdot 8 x^{7} y^{3} + 210 {(x)}^{6} {(2 y)}^{4} + 252 {(x)}^{5} {(2 y)}^{5} + 210 {(x)}^{4} {(2 y)}^{6} + 120 {(x)}^{3} {(2 y)}^{7} + 45 {(x)}^{2} {(2 y)}^{8} + 10 {(x)}^{1} {(2 y)}^{9} + 1 {(x)}^{0} {(2 y)}^{10}$

Step 4.18

Multiply $120$ by $8$ .

$x^{10} + 20 x^{9} y + 180 x^{8} y^{2} + 960 x^{7} y^{3} + 210 {(x)}^{6} {(2 y)}^{4} + 252 {(x)}^{5} {(2 y)}^{5} + 210 {(x)}^{4} {(2 y)}^{6} + 120 {(x)}^{3} {(2 y)}^{7} + 45 {(x)}^{2} {(2 y)}^{8} + 10 {(x)}^{1} {(2 y)}^{9} + 1 {(x)}^{0} {(2 y)}^{10}$

Step 4.19

Apply the product rule to $2 y$ .

$x^{10} + 20 x^{9} y + 180 x^{8} y^{2} + 960 x^{7} y^{3} + 210 x^{6} (2^{4} y^{4}) + 252 {(x)}^{5} {(2 y)}^{5} + 210 {(x)}^{4} {(2 y)}^{6} + 120 {(x)}^{3} {(2 y)}^{7} + 45 {(x)}^{2} {(2 y)}^{8} + 10 {(x)}^{1} {(2 y)}^{9} + 1 {(x)}^{0} {(2 y)}^{10}$

Step 4.20

Rewrite using the commutative property of multiplication.

$x^{10} + 20 x^{9} y + 180 x^{8} y^{2} + 960 x^{7} y^{3} + 210 \cdot 2^{4} x^{6} y^{4} + 252 {(x)}^{5} {(2 y)}^{5} + 210 {(x)}^{4} {(2 y)}^{6} + 120 {(x)}^{3} {(2 y)}^{7} + 45 {(x)}^{2} {(2 y)}^{8} + 10 {(x)}^{1} {(2 y)}^{9} + 1 {(x)}^{0} {(2 y)}^{10}$

Step 4.21

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

$x^{10} + 20 x^{9} y + 180 x^{8} y^{2} + 960 x^{7} y^{3} + 210 \cdot 16 x^{6} y^{4} + 252 {(x)}^{5} {(2 y)}^{5} + 210 {(x)}^{4} {(2 y)}^{6} + 120 {(x)}^{3} {(2 y)}^{7} + 45 {(x)}^{2} {(2 y)}^{8} + 10 {(x)}^{1} {(2 y)}^{9} + 1 {(x)}^{0} {(2 y)}^{10}$

Step 4.22

Multiply $210$ by $16$ .

$x^{10} + 20 x^{9} y + 180 x^{8} y^{2} + 960 x^{7} y^{3} + 3360 x^{6} y^{4} + 252 {(x)}^{5} {(2 y)}^{5} + 210 {(x)}^{4} {(2 y)}^{6} + 120 {(x)}^{3} {(2 y)}^{7} + 45 {(x)}^{2} {(2 y)}^{8} + 10 {(x)}^{1} {(2 y)}^{9} + 1 {(x)}^{0} {(2 y)}^{10}$

Step 4.23

Apply the product rule to $2 y$ .

$x^{10} + 20 x^{9} y + 180 x^{8} y^{2} + 960 x^{7} y^{3} + 3360 x^{6} y^{4} + 252 x^{5} (2^{5} y^{5}) + 210 {(x)}^{4} {(2 y)}^{6} + 120 {(x)}^{3} {(2 y)}^{7} + 45 {(x)}^{2} {(2 y)}^{8} + 10 {(x)}^{1} {(2 y)}^{9} + 1 {(x)}^{0} {(2 y)}^{10}$

Step 4.24

Rewrite using the commutative property of multiplication.

$x^{10} + 20 x^{9} y + 180 x^{8} y^{2} + 960 x^{7} y^{3} + 3360 x^{6} y^{4} + 252 \cdot 2^{5} x^{5} y^{5} + 210 {(x)}^{4} {(2 y)}^{6} + 120 {(x)}^{3} {(2 y)}^{7} + 45 {(x)}^{2} {(2 y)}^{8} + 10 {(x)}^{1} {(2 y)}^{9} + 1 {(x)}^{0} {(2 y)}^{10}$

Step 4.25

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

$x^{10} + 20 x^{9} y + 180 x^{8} y^{2} + 960 x^{7} y^{3} + 3360 x^{6} y^{4} + 252 \cdot 32 x^{5} y^{5} + 210 {(x)}^{4} {(2 y)}^{6} + 120 {(x)}^{3} {(2 y)}^{7} + 45 {(x)}^{2} {(2 y)}^{8} + 10 {(x)}^{1} {(2 y)}^{9} + 1 {(x)}^{0} {(2 y)}^{10}$

Step 4.26

Multiply $252$ by $32$ .

$x^{10} + 20 x^{9} y + 180 x^{8} y^{2} + 960 x^{7} y^{3} + 3360 x^{6} y^{4} + 8064 x^{5} y^{5} + 210 {(x)}^{4} {(2 y)}^{6} + 120 {(x)}^{3} {(2 y)}^{7} + 45 {(x)}^{2} {(2 y)}^{8} + 10 {(x)}^{1} {(2 y)}^{9} + 1 {(x)}^{0} {(2 y)}^{10}$

Step 4.27

Apply the product rule to $2 y$ .

$x^{10} + 20 x^{9} y + 180 x^{8} y^{2} + 960 x^{7} y^{3} + 3360 x^{6} y^{4} + 8064 x^{5} y^{5} + 210 x^{4} (2^{6} y^{6}) + 120 {(x)}^{3} {(2 y)}^{7} + 45 {(x)}^{2} {(2 y)}^{8} + 10 {(x)}^{1} {(2 y)}^{9} + 1 {(x)}^{0} {(2 y)}^{10}$

Step 4.28

Rewrite using the commutative property of multiplication.

$x^{10} + 20 x^{9} y + 180 x^{8} y^{2} + 960 x^{7} y^{3} + 3360 x^{6} y^{4} + 8064 x^{5} y^{5} + 210 \cdot 2^{6} x^{4} y^{6} + 120 {(x)}^{3} {(2 y)}^{7} + 45 {(x)}^{2} {(2 y)}^{8} + 10 {(x)}^{1} {(2 y)}^{9} + 1 {(x)}^{0} {(2 y)}^{10}$

Step 4.29

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

$x^{10} + 20 x^{9} y + 180 x^{8} y^{2} + 960 x^{7} y^{3} + 3360 x^{6} y^{4} + 8064 x^{5} y^{5} + 210 \cdot 64 x^{4} y^{6} + 120 {(x)}^{3} {(2 y)}^{7} + 45 {(x)}^{2} {(2 y)}^{8} + 10 {(x)}^{1} {(2 y)}^{9} + 1 {(x)}^{0} {(2 y)}^{10}$

Step 4.30

Multiply $210$ by $64$ .

$x^{10} + 20 x^{9} y + 180 x^{8} y^{2} + 960 x^{7} y^{3} + 3360 x^{6} y^{4} + 8064 x^{5} y^{5} + 13440 x^{4} y^{6} + 120 {(x)}^{3} {(2 y)}^{7} + 45 {(x)}^{2} {(2 y)}^{8} + 10 {(x)}^{1} {(2 y)}^{9} + 1 {(x)}^{0} {(2 y)}^{10}$

Step 4.31

Apply the product rule to $2 y$ .

$x^{10} + 20 x^{9} y + 180 x^{8} y^{2} + 960 x^{7} y^{3} + 3360 x^{6} y^{4} + 8064 x^{5} y^{5} + 13440 x^{4} y^{6} + 120 x^{3} (2^{7} y^{7}) + 45 {(x)}^{2} {(2 y)}^{8} + 10 {(x)}^{1} {(2 y)}^{9} + 1 {(x)}^{0} {(2 y)}^{10}$

Step 4.32

Rewrite using the commutative property of multiplication.

$x^{10} + 20 x^{9} y + 180 x^{8} y^{2} + 960 x^{7} y^{3} + 3360 x^{6} y^{4} + 8064 x^{5} y^{5} + 13440 x^{4} y^{6} + 120 \cdot 2^{7} x^{3} y^{7} + 45 {(x)}^{2} {(2 y)}^{8} + 10 {(x)}^{1} {(2 y)}^{9} + 1 {(x)}^{0} {(2 y)}^{10}$

Step 4.33

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

$x^{10} + 20 x^{9} y + 180 x^{8} y^{2} + 960 x^{7} y^{3} + 3360 x^{6} y^{4} + 8064 x^{5} y^{5} + 13440 x^{4} y^{6} + 120 \cdot 128 x^{3} y^{7} + 45 {(x)}^{2} {(2 y)}^{8} + 10 {(x)}^{1} {(2 y)}^{9} + 1 {(x)}^{0} {(2 y)}^{10}$

Step 4.34

Multiply $120$ by $128$ .

$x^{10} + 20 x^{9} y + 180 x^{8} y^{2} + 960 x^{7} y^{3} + 3360 x^{6} y^{4} + 8064 x^{5} y^{5} + 13440 x^{4} y^{6} + 15360 x^{3} y^{7} + 45 {(x)}^{2} {(2 y)}^{8} + 10 {(x)}^{1} {(2 y)}^{9} + 1 {(x)}^{0} {(2 y)}^{10}$

Step 4.35

Apply the product rule to $2 y$ .

$x^{10} + 20 x^{9} y + 180 x^{8} y^{2} + 960 x^{7} y^{3} + 3360 x^{6} y^{4} + 8064 x^{5} y^{5} + 13440 x^{4} y^{6} + 15360 x^{3} y^{7} + 45 x^{2} (2^{8} y^{8}) + 10 {(x)}^{1} {(2 y)}^{9} + 1 {(x)}^{0} {(2 y)}^{10}$

Step 4.36

Rewrite using the commutative property of multiplication.

$x^{10} + 20 x^{9} y + 180 x^{8} y^{2} + 960 x^{7} y^{3} + 3360 x^{6} y^{4} + 8064 x^{5} y^{5} + 13440 x^{4} y^{6} + 15360 x^{3} y^{7} + 45 \cdot 2^{8} x^{2} y^{8} + 10 {(x)}^{1} {(2 y)}^{9} + 1 {(x)}^{0} {(2 y)}^{10}$

Step 4.37

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

$x^{10} + 20 x^{9} y + 180 x^{8} y^{2} + 960 x^{7} y^{3} + 3360 x^{6} y^{4} + 8064 x^{5} y^{5} + 13440 x^{4} y^{6} + 15360 x^{3} y^{7} + 45 \cdot 256 x^{2} y^{8} + 10 {(x)}^{1} {(2 y)}^{9} + 1 {(x)}^{0} {(2 y)}^{10}$

Step 4.38

Multiply $45$ by $256$ .

$x^{10} + 20 x^{9} y + 180 x^{8} y^{2} + 960 x^{7} y^{3} + 3360 x^{6} y^{4} + 8064 x^{5} y^{5} + 13440 x^{4} y^{6} + 15360 x^{3} y^{7} + 11520 x^{2} y^{8} + 10 {(x)}^{1} {(2 y)}^{9} + 1 {(x)}^{0} {(2 y)}^{10}$

Step 4.39

Simplify.

$x^{10} + 20 x^{9} y + 180 x^{8} y^{2} + 960 x^{7} y^{3} + 3360 x^{6} y^{4} + 8064 x^{5} y^{5} + 13440 x^{4} y^{6} + 15360 x^{3} y^{7} + 11520 x^{2} y^{8} + 10 x {(2 y)}^{9} + 1 {(x)}^{0} {(2 y)}^{10}$

Step 4.40

Apply the product rule to $2 y$ .

$x^{10} + 20 x^{9} y + 180 x^{8} y^{2} + 960 x^{7} y^{3} + 3360 x^{6} y^{4} + 8064 x^{5} y^{5} + 13440 x^{4} y^{6} + 15360 x^{3} y^{7} + 11520 x^{2} y^{8} + 10 x (2^{9} y^{9}) + 1 {(x)}^{0} {(2 y)}^{10}$

Step 4.41

Rewrite using the commutative property of multiplication.

$x^{10} + 20 x^{9} y + 180 x^{8} y^{2} + 960 x^{7} y^{3} + 3360 x^{6} y^{4} + 8064 x^{5} y^{5} + 13440 x^{4} y^{6} + 15360 x^{3} y^{7} + 11520 x^{2} y^{8} + 10 \cdot 2^{9} x y^{9} + 1 {(x)}^{0} {(2 y)}^{10}$

Step 4.42

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

$x^{10} + 20 x^{9} y + 180 x^{8} y^{2} + 960 x^{7} y^{3} + 3360 x^{6} y^{4} + 8064 x^{5} y^{5} + 13440 x^{4} y^{6} + 15360 x^{3} y^{7} + 11520 x^{2} y^{8} + 10 \cdot 512 x y^{9} + 1 {(x)}^{0} {(2 y)}^{10}$

Step 4.43

Multiply $10$ by $512$ .

$x^{10} + 20 x^{9} y + 180 x^{8} y^{2} + 960 x^{7} y^{3} + 3360 x^{6} y^{4} + 8064 x^{5} y^{5} + 13440 x^{4} y^{6} + 15360 x^{3} y^{7} + 11520 x^{2} y^{8} + 5120 x y^{9} + 1 {(x)}^{0} {(2 y)}^{10}$

Step 4.44

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

$x^{10} + 20 x^{9} y + 180 x^{8} y^{2} + 960 x^{7} y^{3} + 3360 x^{6} y^{4} + 8064 x^{5} y^{5} + 13440 x^{4} y^{6} + 15360 x^{3} y^{7} + 11520 x^{2} y^{8} + 5120 x y^{9} + {(x)}^{0} {(2 y)}^{10}$

Step 4.45

Anything raised to $0$ is $1$ .

$x^{10} + 20 x^{9} y + 180 x^{8} y^{2} + 960 x^{7} y^{3} + 3360 x^{6} y^{4} + 8064 x^{5} y^{5} + 13440 x^{4} y^{6} + 15360 x^{3} y^{7} + 11520 x^{2} y^{8} + 5120 x y^{9} + 1 {(2 y)}^{10}$

Step 4.46

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

$x^{10} + 20 x^{9} y + 180 x^{8} y^{2} + 960 x^{7} y^{3} + 3360 x^{6} y^{4} + 8064 x^{5} y^{5} + 13440 x^{4} y^{6} + 15360 x^{3} y^{7} + 11520 x^{2} y^{8} + 5120 x y^{9} + {(2 y)}^{10}$

Step 4.47

Apply the product rule to $2 y$ .

$x^{10} + 20 x^{9} y + 180 x^{8} y^{2} + 960 x^{7} y^{3} + 3360 x^{6} y^{4} + 8064 x^{5} y^{5} + 13440 x^{4} y^{6} + 15360 x^{3} y^{7} + 11520 x^{2} y^{8} + 5120 x y^{9} + 2^{10} y^{10}$

Step 4.48

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

$x^{10} + 20 x^{9} y + 180 x^{8} y^{2} + 960 x^{7} y^{3} + 3360 x^{6} y^{4} + 8064 x^{5} y^{5} + 13440 x^{4} y^{6} + 15360 x^{3} y^{7} + 11520 x^{2} y^{8} + 5120 x y^{9} + 1024 y^{10}$