Algebra Examples

${(\frac{1}{2} x - y)}^{10}$

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 ${(\frac{1}{2} x - y)}^{10}$ , $n = 10$ 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 $a$ $\frac{1}{2} x$ and $b$ $- y$ into the expression.

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

Step 4

Simplify each term.

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Step 4.1

Multiply ${(\frac{1}{2} x)}^{10}$ by $1$ .

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

Step 4.2

Combine $\frac{1}{2}$ and $x$ .

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

Step 4.3

Apply the product rule to $\frac{x}{2}$ .

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

Step 4.4

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

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

Step 4.5

Apply the product rule to $- y$ .

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

Step 4.6

Rewrite using the commutative property of multiplication.

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

Step 4.7

Anything raised to $0$ is $1$ .

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

Step 4.8

Multiply $\frac{x^{10}}{1024}$ by $1$ .

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

Step 4.9

Anything raised to $0$ is $1$ .

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

Step 4.10

Multiply $\frac{x^{10}}{1024}$ by $1$ .

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

Step 4.11

Combine $\frac{1}{2}$ and $x$ .

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

Step 4.12

Apply the product rule to $\frac{x}{2}$ .

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

Step 4.13

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

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

Step 4.14

Cancel the common factor of $2$ .

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Step 4.14.1

Factor $2$ out of $10$ .

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

Step 4.14.2

Factor $2$ out of $512$ .

$\frac{x^{10}}{1024} + 2 \cdot 5 \frac{x^{9}}{2 \cdot 256} {(- y)}^{1} + 45 {(\frac{1}{2} x)}^{8} {(- y)}^{2} + 120 {(\frac{1}{2} x)}^{7} {(- y)}^{3} + 210 {(\frac{1}{2} x)}^{6} {(- y)}^{4} + 252 {(\frac{1}{2} x)}^{5} {(- y)}^{5} + 210 {(\frac{1}{2} x)}^{4} {(- y)}^{6} + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.14.3

Cancel the common factor.

Step 4.14.4

Rewrite the expression.

$\frac{x^{10}}{1024} + 5 \frac{x^{9}}{256} {(- y)}^{1} + 45 {(\frac{1}{2} x)}^{8} {(- y)}^{2} + 120 {(\frac{1}{2} x)}^{7} {(- y)}^{3} + 210 {(\frac{1}{2} x)}^{6} {(- y)}^{4} + 252 {(\frac{1}{2} x)}^{5} {(- y)}^{5} + 210 {(\frac{1}{2} x)}^{4} {(- y)}^{6} + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.15

Combine $5$ and $\frac{x^{9}}{256}$ .

$\frac{x^{10}}{1024} + \frac{5 x^{9}}{256} {(- y)}^{1} + 45 {(\frac{1}{2} x)}^{8} {(- y)}^{2} + 120 {(\frac{1}{2} x)}^{7} {(- y)}^{3} + 210 {(\frac{1}{2} x)}^{6} {(- y)}^{4} + 252 {(\frac{1}{2} x)}^{5} {(- y)}^{5} + 210 {(\frac{1}{2} x)}^{4} {(- y)}^{6} + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.16

Simplify.

$\frac{x^{10}}{1024} + \frac{5 x^{9}}{256} (- y) + 45 {(\frac{1}{2} x)}^{8} {(- y)}^{2} + 120 {(\frac{1}{2} x)}^{7} {(- y)}^{3} + 210 {(\frac{1}{2} x)}^{6} {(- y)}^{4} + 252 {(\frac{1}{2} x)}^{5} {(- y)}^{5} + 210 {(\frac{1}{2} x)}^{4} {(- y)}^{6} + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.17

Rewrite using the commutative property of multiplication.

$\frac{x^{10}}{1024} - \frac{5 x^{9}}{256} y + 45 {(\frac{1}{2} x)}^{8} {(- y)}^{2} + 120 {(\frac{1}{2} x)}^{7} {(- y)}^{3} + 210 {(\frac{1}{2} x)}^{6} {(- y)}^{4} + 252 {(\frac{1}{2} x)}^{5} {(- y)}^{5} + 210 {(\frac{1}{2} x)}^{4} {(- y)}^{6} + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.18

Combine $y$ and $\frac{5 x^{9}}{256}$ .

$\frac{x^{10}}{1024} - \frac{y (5 x^{9})}{256} + 45 {(\frac{1}{2} x)}^{8} {(- y)}^{2} + 120 {(\frac{1}{2} x)}^{7} {(- y)}^{3} + 210 {(\frac{1}{2} x)}^{6} {(- y)}^{4} + 252 {(\frac{1}{2} x)}^{5} {(- y)}^{5} + 210 {(\frac{1}{2} x)}^{4} {(- y)}^{6} + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.19

Move $5$ to the left of $y$ .

$\frac{x^{10}}{1024} - \frac{5 \cdot y x^{9}}{256} + 45 {(\frac{1}{2} x)}^{8} {(- y)}^{2} + 120 {(\frac{1}{2} x)}^{7} {(- y)}^{3} + 210 {(\frac{1}{2} x)}^{6} {(- y)}^{4} + 252 {(\frac{1}{2} x)}^{5} {(- y)}^{5} + 210 {(\frac{1}{2} x)}^{4} {(- y)}^{6} + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.20

Combine $\frac{1}{2}$ and $x$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + 45 {(\frac{x}{2})}^{8} {(- y)}^{2} + 120 {(\frac{1}{2} x)}^{7} {(- y)}^{3} + 210 {(\frac{1}{2} x)}^{6} {(- y)}^{4} + 252 {(\frac{1}{2} x)}^{5} {(- y)}^{5} + 210 {(\frac{1}{2} x)}^{4} {(- y)}^{6} + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.21

Apply the product rule to $\frac{x}{2}$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + 45 \frac{x^{8}}{2^{8}} {(- y)}^{2} + 120 {(\frac{1}{2} x)}^{7} {(- y)}^{3} + 210 {(\frac{1}{2} x)}^{6} {(- y)}^{4} + 252 {(\frac{1}{2} x)}^{5} {(- y)}^{5} + 210 {(\frac{1}{2} x)}^{4} {(- y)}^{6} + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.22

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

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + 45 \frac{x^{8}}{256} {(- y)}^{2} + 120 {(\frac{1}{2} x)}^{7} {(- y)}^{3} + 210 {(\frac{1}{2} x)}^{6} {(- y)}^{4} + 252 {(\frac{1}{2} x)}^{5} {(- y)}^{5} + 210 {(\frac{1}{2} x)}^{4} {(- y)}^{6} + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.23

Combine $45$ and $\frac{x^{8}}{256}$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8}}{256} {(- y)}^{2} + 120 {(\frac{1}{2} x)}^{7} {(- y)}^{3} + 210 {(\frac{1}{2} x)}^{6} {(- y)}^{4} + 252 {(\frac{1}{2} x)}^{5} {(- y)}^{5} + 210 {(\frac{1}{2} x)}^{4} {(- y)}^{6} + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.24

Apply the product rule to $- y$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8}}{256} ({(- 1)}^{2} y^{2}) + 120 {(\frac{1}{2} x)}^{7} {(- y)}^{3} + 210 {(\frac{1}{2} x)}^{6} {(- y)}^{4} + 252 {(\frac{1}{2} x)}^{5} {(- y)}^{5} + 210 {(\frac{1}{2} x)}^{4} {(- y)}^{6} + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.25

Rewrite using the commutative property of multiplication.

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + {(- 1)}^{2} \frac{45 x^{8}}{256} y^{2} + 120 {(\frac{1}{2} x)}^{7} {(- y)}^{3} + 210 {(\frac{1}{2} x)}^{6} {(- y)}^{4} + 252 {(\frac{1}{2} x)}^{5} {(- y)}^{5} + 210 {(\frac{1}{2} x)}^{4} {(- y)}^{6} + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.26

Raise $- 1$ to the power of $2$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + 1 \frac{45 x^{8}}{256} y^{2} + 120 {(\frac{1}{2} x)}^{7} {(- y)}^{3} + 210 {(\frac{1}{2} x)}^{6} {(- y)}^{4} + 252 {(\frac{1}{2} x)}^{5} {(- y)}^{5} + 210 {(\frac{1}{2} x)}^{4} {(- y)}^{6} + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.27

Multiply $\frac{45 x^{8}}{256}$ by $1$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8}}{256} y^{2} + 120 {(\frac{1}{2} x)}^{7} {(- y)}^{3} + 210 {(\frac{1}{2} x)}^{6} {(- y)}^{4} + 252 {(\frac{1}{2} x)}^{5} {(- y)}^{5} + 210 {(\frac{1}{2} x)}^{4} {(- y)}^{6} + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.28

Combine $\frac{45 x^{8}}{256}$ and $y^{2}$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} + 120 {(\frac{1}{2} x)}^{7} {(- y)}^{3} + 210 {(\frac{1}{2} x)}^{6} {(- y)}^{4} + 252 {(\frac{1}{2} x)}^{5} {(- y)}^{5} + 210 {(\frac{1}{2} x)}^{4} {(- y)}^{6} + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.29

Combine $\frac{1}{2}$ and $x$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} + 120 {(\frac{x}{2})}^{7} {(- y)}^{3} + 210 {(\frac{1}{2} x)}^{6} {(- y)}^{4} + 252 {(\frac{1}{2} x)}^{5} {(- y)}^{5} + 210 {(\frac{1}{2} x)}^{4} {(- y)}^{6} + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.30

Apply the product rule to $\frac{x}{2}$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} + 120 \frac{x^{7}}{2^{7}} {(- y)}^{3} + 210 {(\frac{1}{2} x)}^{6} {(- y)}^{4} + 252 {(\frac{1}{2} x)}^{5} {(- y)}^{5} + 210 {(\frac{1}{2} x)}^{4} {(- y)}^{6} + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.31

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

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} + 120 \frac{x^{7}}{128} {(- y)}^{3} + 210 {(\frac{1}{2} x)}^{6} {(- y)}^{4} + 252 {(\frac{1}{2} x)}^{5} {(- y)}^{5} + 210 {(\frac{1}{2} x)}^{4} {(- y)}^{6} + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.32

Cancel the common factor of $8$ .

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Step 4.32.1

Factor $8$ out of $120$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} + 8 (15) \frac{x^{7}}{128} {(- y)}^{3} + 210 {(\frac{1}{2} x)}^{6} {(- y)}^{4} + 252 {(\frac{1}{2} x)}^{5} {(- y)}^{5} + 210 {(\frac{1}{2} x)}^{4} {(- y)}^{6} + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.32.2

Factor $8$ out of $128$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} + 8 \cdot 15 \frac{x^{7}}{8 \cdot 16} {(- y)}^{3} + 210 {(\frac{1}{2} x)}^{6} {(- y)}^{4} + 252 {(\frac{1}{2} x)}^{5} {(- y)}^{5} + 210 {(\frac{1}{2} x)}^{4} {(- y)}^{6} + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.32.3

Cancel the common factor.

Step 4.32.4

Rewrite the expression.

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} + 15 \frac{x^{7}}{16} {(- y)}^{3} + 210 {(\frac{1}{2} x)}^{6} {(- y)}^{4} + 252 {(\frac{1}{2} x)}^{5} {(- y)}^{5} + 210 {(\frac{1}{2} x)}^{4} {(- y)}^{6} + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.33

Combine $15$ and $\frac{x^{7}}{16}$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} + \frac{15 x^{7}}{16} {(- y)}^{3} + 210 {(\frac{1}{2} x)}^{6} {(- y)}^{4} + 252 {(\frac{1}{2} x)}^{5} {(- y)}^{5} + 210 {(\frac{1}{2} x)}^{4} {(- y)}^{6} + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.34

Apply the product rule to $- y$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} + \frac{15 x^{7}}{16} ({(- 1)}^{3} y^{3}) + 210 {(\frac{1}{2} x)}^{6} {(- y)}^{4} + 252 {(\frac{1}{2} x)}^{5} {(- y)}^{5} + 210 {(\frac{1}{2} x)}^{4} {(- y)}^{6} + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.35

Rewrite using the commutative property of multiplication.

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} + {(- 1)}^{3} \frac{15 x^{7}}{16} y^{3} + 210 {(\frac{1}{2} x)}^{6} {(- y)}^{4} + 252 {(\frac{1}{2} x)}^{5} {(- y)}^{5} + 210 {(\frac{1}{2} x)}^{4} {(- y)}^{6} + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.36

Raise $- 1$ to the power of $3$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 x^{7}}{16} y^{3} + 210 {(\frac{1}{2} x)}^{6} {(- y)}^{4} + 252 {(\frac{1}{2} x)}^{5} {(- y)}^{5} + 210 {(\frac{1}{2} x)}^{4} {(- y)}^{6} + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.37

Combine $y^{3}$ and $\frac{15 x^{7}}{16}$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{y^{3} (15 x^{7})}{16} + 210 {(\frac{1}{2} x)}^{6} {(- y)}^{4} + 252 {(\frac{1}{2} x)}^{5} {(- y)}^{5} + 210 {(\frac{1}{2} x)}^{4} {(- y)}^{6} + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.38

Move $15$ to the left of $y^{3}$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 \cdot y^{3} x^{7}}{16} + 210 {(\frac{1}{2} x)}^{6} {(- y)}^{4} + 252 {(\frac{1}{2} x)}^{5} {(- y)}^{5} + 210 {(\frac{1}{2} x)}^{4} {(- y)}^{6} + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.39

Combine $\frac{1}{2}$ and $x$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + 210 {(\frac{x}{2})}^{6} {(- y)}^{4} + 252 {(\frac{1}{2} x)}^{5} {(- y)}^{5} + 210 {(\frac{1}{2} x)}^{4} {(- y)}^{6} + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.40

Apply the product rule to $\frac{x}{2}$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + 210 \frac{x^{6}}{2^{6}} {(- y)}^{4} + 252 {(\frac{1}{2} x)}^{5} {(- y)}^{5} + 210 {(\frac{1}{2} x)}^{4} {(- y)}^{6} + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.41

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

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + 210 \frac{x^{6}}{64} {(- y)}^{4} + 252 {(\frac{1}{2} x)}^{5} {(- y)}^{5} + 210 {(\frac{1}{2} x)}^{4} {(- y)}^{6} + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.42

Cancel the common factor of $2$ .

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Step 4.42.1

Factor $2$ out of $210$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + 2 (105) \frac{x^{6}}{64} {(- y)}^{4} + 252 {(\frac{1}{2} x)}^{5} {(- y)}^{5} + 210 {(\frac{1}{2} x)}^{4} {(- y)}^{6} + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.42.2

Factor $2$ out of $64$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + 2 \cdot 105 \frac{x^{6}}{2 \cdot 32} {(- y)}^{4} + 252 {(\frac{1}{2} x)}^{5} {(- y)}^{5} + 210 {(\frac{1}{2} x)}^{4} {(- y)}^{6} + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.42.3

Cancel the common factor.

Step 4.42.4

Rewrite the expression.

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + 105 \frac{x^{6}}{32} {(- y)}^{4} + 252 {(\frac{1}{2} x)}^{5} {(- y)}^{5} + 210 {(\frac{1}{2} x)}^{4} {(- y)}^{6} + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.43

Combine $105$ and $\frac{x^{6}}{32}$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6}}{32} {(- y)}^{4} + 252 {(\frac{1}{2} x)}^{5} {(- y)}^{5} + 210 {(\frac{1}{2} x)}^{4} {(- y)}^{6} + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.44

Apply the product rule to $- y$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6}}{32} ({(- 1)}^{4} y^{4}) + 252 {(\frac{1}{2} x)}^{5} {(- y)}^{5} + 210 {(\frac{1}{2} x)}^{4} {(- y)}^{6} + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.45

Rewrite using the commutative property of multiplication.

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + {(- 1)}^{4} \frac{105 x^{6}}{32} y^{4} + 252 {(\frac{1}{2} x)}^{5} {(- y)}^{5} + 210 {(\frac{1}{2} x)}^{4} {(- y)}^{6} + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.46

Raise $- 1$ to the power of $4$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + 1 \frac{105 x^{6}}{32} y^{4} + 252 {(\frac{1}{2} x)}^{5} {(- y)}^{5} + 210 {(\frac{1}{2} x)}^{4} {(- y)}^{6} + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.47

Multiply $\frac{105 x^{6}}{32}$ by $1$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6}}{32} y^{4} + 252 {(\frac{1}{2} x)}^{5} {(- y)}^{5} + 210 {(\frac{1}{2} x)}^{4} {(- y)}^{6} + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.48

Combine $\frac{105 x^{6}}{32}$ and $y^{4}$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6} y^{4}}{32} + 252 {(\frac{1}{2} x)}^{5} {(- y)}^{5} + 210 {(\frac{1}{2} x)}^{4} {(- y)}^{6} + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.49

Combine $\frac{1}{2}$ and $x$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6} y^{4}}{32} + 252 {(\frac{x}{2})}^{5} {(- y)}^{5} + 210 {(\frac{1}{2} x)}^{4} {(- y)}^{6} + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.50

Apply the product rule to $\frac{x}{2}$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6} y^{4}}{32} + 252 \frac{x^{5}}{2^{5}} {(- y)}^{5} + 210 {(\frac{1}{2} x)}^{4} {(- y)}^{6} + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.51

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

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6} y^{4}}{32} + 252 \frac{x^{5}}{32} {(- y)}^{5} + 210 {(\frac{1}{2} x)}^{4} {(- y)}^{6} + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.52

Cancel the common factor of $4$ .

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Step 4.52.1

Factor $4$ out of $252$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6} y^{4}}{32} + 4 (63) \frac{x^{5}}{32} {(- y)}^{5} + 210 {(\frac{1}{2} x)}^{4} {(- y)}^{6} + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.52.2

Factor $4$ out of $32$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6} y^{4}}{32} + 4 \cdot 63 \frac{x^{5}}{4 \cdot 8} {(- y)}^{5} + 210 {(\frac{1}{2} x)}^{4} {(- y)}^{6} + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.52.3

Cancel the common factor.

Step 4.52.4

Rewrite the expression.

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6} y^{4}}{32} + 63 \frac{x^{5}}{8} {(- y)}^{5} + 210 {(\frac{1}{2} x)}^{4} {(- y)}^{6} + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.53

Combine $63$ and $\frac{x^{5}}{8}$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6} y^{4}}{32} + \frac{63 x^{5}}{8} {(- y)}^{5} + 210 {(\frac{1}{2} x)}^{4} {(- y)}^{6} + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.54

Apply the product rule to $- y$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6} y^{4}}{32} + \frac{63 x^{5}}{8} ({(- 1)}^{5} y^{5}) + 210 {(\frac{1}{2} x)}^{4} {(- y)}^{6} + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.55

Rewrite using the commutative property of multiplication.

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6} y^{4}}{32} + {(- 1)}^{5} \frac{63 x^{5}}{8} y^{5} + 210 {(\frac{1}{2} x)}^{4} {(- y)}^{6} + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.56

Raise $- 1$ to the power of $5$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6} y^{4}}{32} - \frac{63 x^{5}}{8} y^{5} + 210 {(\frac{1}{2} x)}^{4} {(- y)}^{6} + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.57

Combine $y^{5}$ and $\frac{63 x^{5}}{8}$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6} y^{4}}{32} - \frac{y^{5} (63 x^{5})}{8} + 210 {(\frac{1}{2} x)}^{4} {(- y)}^{6} + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.58

Move $63$ to the left of $y^{5}$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6} y^{4}}{32} - \frac{63 \cdot y^{5} x^{5}}{8} + 210 {(\frac{1}{2} x)}^{4} {(- y)}^{6} + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.59

Combine $\frac{1}{2}$ and $x$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6} y^{4}}{32} - \frac{63 y^{5} x^{5}}{8} + 210 {(\frac{x}{2})}^{4} {(- y)}^{6} + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.60

Apply the product rule to $\frac{x}{2}$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6} y^{4}}{32} - \frac{63 y^{5} x^{5}}{8} + 210 \frac{x^{4}}{2^{4}} {(- y)}^{6} + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.61

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

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6} y^{4}}{32} - \frac{63 y^{5} x^{5}}{8} + 210 \frac{x^{4}}{16} {(- y)}^{6} + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.62

Cancel the common factor of $2$ .

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Step 4.62.1

Factor $2$ out of $210$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6} y^{4}}{32} - \frac{63 y^{5} x^{5}}{8} + 2 (105) \frac{x^{4}}{16} {(- y)}^{6} + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.62.2

Factor $2$ out of $16$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6} y^{4}}{32} - \frac{63 y^{5} x^{5}}{8} + 2 \cdot 105 \frac{x^{4}}{2 \cdot 8} {(- y)}^{6} + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.62.3

Cancel the common factor.

Step 4.62.4

Rewrite the expression.

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6} y^{4}}{32} - \frac{63 y^{5} x^{5}}{8} + 105 \frac{x^{4}}{8} {(- y)}^{6} + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.63

Combine $105$ and $\frac{x^{4}}{8}$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6} y^{4}}{32} - \frac{63 y^{5} x^{5}}{8} + \frac{105 x^{4}}{8} {(- y)}^{6} + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.64

Apply the product rule to $- y$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6} y^{4}}{32} - \frac{63 y^{5} x^{5}}{8} + \frac{105 x^{4}}{8} ({(- 1)}^{6} y^{6}) + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.65

Rewrite using the commutative property of multiplication.

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6} y^{4}}{32} - \frac{63 y^{5} x^{5}}{8} + {(- 1)}^{6} \frac{105 x^{4}}{8} y^{6} + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.66

Raise $- 1$ to the power of $6$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6} y^{4}}{32} - \frac{63 y^{5} x^{5}}{8} + 1 \frac{105 x^{4}}{8} y^{6} + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.67

Multiply $\frac{105 x^{4}}{8}$ by $1$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6} y^{4}}{32} - \frac{63 y^{5} x^{5}}{8} + \frac{105 x^{4}}{8} y^{6} + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.68

Combine $\frac{105 x^{4}}{8}$ and $y^{6}$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6} y^{4}}{32} - \frac{63 y^{5} x^{5}}{8} + \frac{105 x^{4} y^{6}}{8} + 120 {(\frac{1}{2} x)}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.69

Combine $\frac{1}{2}$ and $x$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6} y^{4}}{32} - \frac{63 y^{5} x^{5}}{8} + \frac{105 x^{4} y^{6}}{8} + 120 {(\frac{x}{2})}^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.70

Apply the product rule to $\frac{x}{2}$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6} y^{4}}{32} - \frac{63 y^{5} x^{5}}{8} + \frac{105 x^{4} y^{6}}{8} + 120 \frac{x^{3}}{2^{3}} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.71

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

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6} y^{4}}{32} - \frac{63 y^{5} x^{5}}{8} + \frac{105 x^{4} y^{6}}{8} + 120 \frac{x^{3}}{8} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.72

Cancel the common factor of $8$ .

Tap for more steps...

Step 4.72.1

Factor $8$ out of $120$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6} y^{4}}{32} - \frac{63 y^{5} x^{5}}{8} + \frac{105 x^{4} y^{6}}{8} + 8 (15) \frac{x^{3}}{8} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.72.2

Cancel the common factor.

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6} y^{4}}{32} - \frac{63 y^{5} x^{5}}{8} + \frac{105 x^{4} y^{6}}{8} + 8 \cdot 15 \frac{x^{3}}{8} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.72.3

Rewrite the expression.

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6} y^{4}}{32} - \frac{63 y^{5} x^{5}}{8} + \frac{105 x^{4} y^{6}}{8} + 15 x^{3} {(- y)}^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.73

Apply the product rule to $- y$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6} y^{4}}{32} - \frac{63 y^{5} x^{5}}{8} + \frac{105 x^{4} y^{6}}{8} + 15 x^{3} ({(- 1)}^{7} y^{7}) + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.74

Rewrite using the commutative property of multiplication.

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6} y^{4}}{32} - \frac{63 y^{5} x^{5}}{8} + \frac{105 x^{4} y^{6}}{8} + 15 \cdot {(- 1)}^{7} x^{3} y^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.75

Raise $- 1$ to the power of $7$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6} y^{4}}{32} - \frac{63 y^{5} x^{5}}{8} + \frac{105 x^{4} y^{6}}{8} + 15 \cdot - 1 x^{3} y^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.76

Multiply $15$ by $- 1$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6} y^{4}}{32} - \frac{63 y^{5} x^{5}}{8} + \frac{105 x^{4} y^{6}}{8} - 15 x^{3} y^{7} + 45 {(\frac{1}{2} x)}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.77

Combine $\frac{1}{2}$ and $x$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6} y^{4}}{32} - \frac{63 y^{5} x^{5}}{8} + \frac{105 x^{4} y^{6}}{8} - 15 x^{3} y^{7} + 45 {(\frac{x}{2})}^{2} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.78

Apply the product rule to $\frac{x}{2}$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6} y^{4}}{32} - \frac{63 y^{5} x^{5}}{8} + \frac{105 x^{4} y^{6}}{8} - 15 x^{3} y^{7} + 45 \frac{x^{2}}{2^{2}} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.79

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

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6} y^{4}}{32} - \frac{63 y^{5} x^{5}}{8} + \frac{105 x^{4} y^{6}}{8} - 15 x^{3} y^{7} + 45 \frac{x^{2}}{4} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.80

Combine $45$ and $\frac{x^{2}}{4}$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6} y^{4}}{32} - \frac{63 y^{5} x^{5}}{8} + \frac{105 x^{4} y^{6}}{8} - 15 x^{3} y^{7} + \frac{45 x^{2}}{4} {(- y)}^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.81

Apply the product rule to $- y$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6} y^{4}}{32} - \frac{63 y^{5} x^{5}}{8} + \frac{105 x^{4} y^{6}}{8} - 15 x^{3} y^{7} + \frac{45 x^{2}}{4} ({(- 1)}^{8} y^{8}) + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.82

Rewrite using the commutative property of multiplication.

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6} y^{4}}{32} - \frac{63 y^{5} x^{5}}{8} + \frac{105 x^{4} y^{6}}{8} - 15 x^{3} y^{7} + {(- 1)}^{8} \frac{45 x^{2}}{4} y^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.83

Raise $- 1$ to the power of $8$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6} y^{4}}{32} - \frac{63 y^{5} x^{5}}{8} + \frac{105 x^{4} y^{6}}{8} - 15 x^{3} y^{7} + 1 \frac{45 x^{2}}{4} y^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.84

Multiply $\frac{45 x^{2}}{4}$ by $1$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6} y^{4}}{32} - \frac{63 y^{5} x^{5}}{8} + \frac{105 x^{4} y^{6}}{8} - 15 x^{3} y^{7} + \frac{45 x^{2}}{4} y^{8} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.85

Combine $\frac{45 x^{2}}{4}$ and $y^{8}$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6} y^{4}}{32} - \frac{63 y^{5} x^{5}}{8} + \frac{105 x^{4} y^{6}}{8} - 15 x^{3} y^{7} + \frac{45 x^{2} y^{8}}{4} + 10 {(\frac{1}{2} x)}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.86

Combine $\frac{1}{2}$ and $x$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6} y^{4}}{32} - \frac{63 y^{5} x^{5}}{8} + \frac{105 x^{4} y^{6}}{8} - 15 x^{3} y^{7} + \frac{45 x^{2} y^{8}}{4} + 10 {(\frac{x}{2})}^{1} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.87

Simplify.

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6} y^{4}}{32} - \frac{63 y^{5} x^{5}}{8} + \frac{105 x^{4} y^{6}}{8} - 15 x^{3} y^{7} + \frac{45 x^{2} y^{8}}{4} + 10 \frac{x}{2} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.88

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.88.1

Factor $2$ out of $10$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6} y^{4}}{32} - \frac{63 y^{5} x^{5}}{8} + \frac{105 x^{4} y^{6}}{8} - 15 x^{3} y^{7} + \frac{45 x^{2} y^{8}}{4} + 2 (5) \frac{x}{2} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.88.2

Cancel the common factor.

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6} y^{4}}{32} - \frac{63 y^{5} x^{5}}{8} + \frac{105 x^{4} y^{6}}{8} - 15 x^{3} y^{7} + \frac{45 x^{2} y^{8}}{4} + 2 \cdot 5 \frac{x}{2} {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.88.3

Rewrite the expression.

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6} y^{4}}{32} - \frac{63 y^{5} x^{5}}{8} + \frac{105 x^{4} y^{6}}{8} - 15 x^{3} y^{7} + \frac{45 x^{2} y^{8}}{4} + 5 x {(- y)}^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.89

Apply the product rule to $- y$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6} y^{4}}{32} - \frac{63 y^{5} x^{5}}{8} + \frac{105 x^{4} y^{6}}{8} - 15 x^{3} y^{7} + \frac{45 x^{2} y^{8}}{4} + 5 x ({(- 1)}^{9} y^{9}) + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.90

Rewrite using the commutative property of multiplication.

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6} y^{4}}{32} - \frac{63 y^{5} x^{5}}{8} + \frac{105 x^{4} y^{6}}{8} - 15 x^{3} y^{7} + \frac{45 x^{2} y^{8}}{4} + 5 \cdot {(- 1)}^{9} x y^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.91

Raise $- 1$ to the power of $9$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6} y^{4}}{32} - \frac{63 y^{5} x^{5}}{8} + \frac{105 x^{4} y^{6}}{8} - 15 x^{3} y^{7} + \frac{45 x^{2} y^{8}}{4} + 5 \cdot - 1 x y^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.92

Multiply $5$ by $- 1$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6} y^{4}}{32} - \frac{63 y^{5} x^{5}}{8} + \frac{105 x^{4} y^{6}}{8} - 15 x^{3} y^{7} + \frac{45 x^{2} y^{8}}{4} - 5 x y^{9} + 1 {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.93

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

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6} y^{4}}{32} - \frac{63 y^{5} x^{5}}{8} + \frac{105 x^{4} y^{6}}{8} - 15 x^{3} y^{7} + \frac{45 x^{2} y^{8}}{4} - 5 x y^{9} + {(\frac{1}{2} x)}^{0} {(- y)}^{10}$

Step 4.94

Combine $\frac{1}{2}$ and $x$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6} y^{4}}{32} - \frac{63 y^{5} x^{5}}{8} + \frac{105 x^{4} y^{6}}{8} - 15 x^{3} y^{7} + \frac{45 x^{2} y^{8}}{4} - 5 x y^{9} + {(\frac{x}{2})}^{0} {(- y)}^{10}$

Step 4.95

Apply the product rule to $\frac{x}{2}$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6} y^{4}}{32} - \frac{63 y^{5} x^{5}}{8} + \frac{105 x^{4} y^{6}}{8} - 15 x^{3} y^{7} + \frac{45 x^{2} y^{8}}{4} - 5 x y^{9} + \frac{x^{0}}{2^{0}} {(- y)}^{10}$

Step 4.96

Anything raised to $0$ is $1$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6} y^{4}}{32} - \frac{63 y^{5} x^{5}}{8} + \frac{105 x^{4} y^{6}}{8} - 15 x^{3} y^{7} + \frac{45 x^{2} y^{8}}{4} - 5 x y^{9} + \frac{1}{2^{0}} {(- y)}^{10}$

Step 4.97

Anything raised to $0$ is $1$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6} y^{4}}{32} - \frac{63 y^{5} x^{5}}{8} + \frac{105 x^{4} y^{6}}{8} - 15 x^{3} y^{7} + \frac{45 x^{2} y^{8}}{4} - 5 x y^{9} + \frac{1}{1} {(- y)}^{10}$

Step 4.98

Divide $1$ by $1$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6} y^{4}}{32} - \frac{63 y^{5} x^{5}}{8} + \frac{105 x^{4} y^{6}}{8} - 15 x^{3} y^{7} + \frac{45 x^{2} y^{8}}{4} - 5 x y^{9} + 1 {(- y)}^{10}$

Step 4.99

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

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6} y^{4}}{32} - \frac{63 y^{5} x^{5}}{8} + \frac{105 x^{4} y^{6}}{8} - 15 x^{3} y^{7} + \frac{45 x^{2} y^{8}}{4} - 5 x y^{9} + {(- y)}^{10}$

Step 4.100

Apply the product rule to $- y$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6} y^{4}}{32} - \frac{63 y^{5} x^{5}}{8} + \frac{105 x^{4} y^{6}}{8} - 15 x^{3} y^{7} + \frac{45 x^{2} y^{8}}{4} - 5 x y^{9} + {(- 1)}^{10} y^{10}$

Step 4.101

Raise $- 1$ to the power of $10$ .

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6} y^{4}}{32} - \frac{63 y^{5} x^{5}}{8} + \frac{105 x^{4} y^{6}}{8} - 15 x^{3} y^{7} + \frac{45 x^{2} y^{8}}{4} - 5 x y^{9} + 1 y^{10}$

Step 4.102

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

$\frac{x^{10}}{1024} - \frac{5 y x^{9}}{256} + \frac{45 x^{8} y^{2}}{256} - \frac{15 y^{3} x^{7}}{16} + \frac{105 x^{6} y^{4}}{32} - \frac{63 y^{5} x^{5}}{8} + \frac{105 x^{4} y^{6}}{8} - 15 x^{3} y^{7} + \frac{45 x^{2} y^{8}}{4} - 5 x y^{9} + y^{10}$