Algebra Examples

${(- x + 5 y)}^{8}$

Step 1

Pascal's Triangle can be displayed as such:

$1$

$1 - 1$

$1 - 2 - 1$

$\dots$

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

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

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

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 ${(- x + 5 y)}^{8}$ , $n = 8$ so the coefficients of the expansion will correspond with line $9$ .

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

$1 a^{8} b^{0} + 8 a^{7} b + 28 a^{6} b^{2} + 56 a^{5} b^{3} + 70 a^{4} b^{4} + 56 a^{3} b^{5} + 28 a^{2} b^{6} + 8 a b^{7} + 1 a^{0} b^{8}$

Step 3

Substitute the actual values of $a$ $- x$ and $b$ $5 y$ into the expression.

$1 {(- x)}^{8} {(5 y)}^{0} + 8 {(- x)}^{7} {(5 y)}^{1} + 28 {(- x)}^{6} {(5 y)}^{2} + 56 {(- x)}^{5} {(5 y)}^{3} + 70 {(- x)}^{4} {(5 y)}^{4} + 56 {(- x)}^{3} {(5 y)}^{5} + 28 {(- x)}^{2} {(5 y)}^{6} + 8 {(- x)}^{1} {(5 y)}^{7} + 1 {(- x)}^{0} {(5 y)}^{8}$

Step 4

Simplify each term.

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Step 4.1

Multiply ${(- x)}^{8}$ by $1$ .

${(- x)}^{8} {(5 y)}^{0} + 8 {(- x)}^{7} {(5 y)}^{1} + 28 {(- x)}^{6} {(5 y)}^{2} + 56 {(- x)}^{5} {(5 y)}^{3} + 70 {(- x)}^{4} {(5 y)}^{4} + 56 {(- x)}^{3} {(5 y)}^{5} + 28 {(- x)}^{2} {(5 y)}^{6} + 8 {(- x)}^{1} {(5 y)}^{7} + 1 {(- x)}^{0} {(5 y)}^{8}$

Step 4.2

Apply the product rule to $- x$ .

${(- 1)}^{8} x^{8} {(5 y)}^{0} + 8 {(- x)}^{7} {(5 y)}^{1} + 28 {(- x)}^{6} {(5 y)}^{2} + 56 {(- x)}^{5} {(5 y)}^{3} + 70 {(- x)}^{4} {(5 y)}^{4} + 56 {(- x)}^{3} {(5 y)}^{5} + 28 {(- x)}^{2} {(5 y)}^{6} + 8 {(- x)}^{1} {(5 y)}^{7} + 1 {(- x)}^{0} {(5 y)}^{8}$

Step 4.3

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

$1 x^{8} {(5 y)}^{0} + 8 {(- x)}^{7} {(5 y)}^{1} + 28 {(- x)}^{6} {(5 y)}^{2} + 56 {(- x)}^{5} {(5 y)}^{3} + 70 {(- x)}^{4} {(5 y)}^{4} + 56 {(- x)}^{3} {(5 y)}^{5} + 28 {(- x)}^{2} {(5 y)}^{6} + 8 {(- x)}^{1} {(5 y)}^{7} + 1 {(- x)}^{0} {(5 y)}^{8}$

Step 4.4

Multiply $x^{8}$ by $1$ .

$x^{8} {(5 y)}^{0} + 8 {(- x)}^{7} {(5 y)}^{1} + 28 {(- x)}^{6} {(5 y)}^{2} + 56 {(- x)}^{5} {(5 y)}^{3} + 70 {(- x)}^{4} {(5 y)}^{4} + 56 {(- x)}^{3} {(5 y)}^{5} + 28 {(- x)}^{2} {(5 y)}^{6} + 8 {(- x)}^{1} {(5 y)}^{7} + 1 {(- x)}^{0} {(5 y)}^{8}$

Step 4.5

Apply the product rule to $5 y$ .

$x^{8} (5^{0} y^{0}) + 8 {(- x)}^{7} {(5 y)}^{1} + 28 {(- x)}^{6} {(5 y)}^{2} + 56 {(- x)}^{5} {(5 y)}^{3} + 70 {(- x)}^{4} {(5 y)}^{4} + 56 {(- x)}^{3} {(5 y)}^{5} + 28 {(- x)}^{2} {(5 y)}^{6} + 8 {(- x)}^{1} {(5 y)}^{7} + 1 {(- x)}^{0} {(5 y)}^{8}$

Step 4.6

Rewrite using the commutative property of multiplication.

$5^{0} x^{8} y^{0} + 8 {(- x)}^{7} {(5 y)}^{1} + 28 {(- x)}^{6} {(5 y)}^{2} + 56 {(- x)}^{5} {(5 y)}^{3} + 70 {(- x)}^{4} {(5 y)}^{4} + 56 {(- x)}^{3} {(5 y)}^{5} + 28 {(- x)}^{2} {(5 y)}^{6} + 8 {(- x)}^{1} {(5 y)}^{7} + 1 {(- x)}^{0} {(5 y)}^{8}$

Step 4.7

Anything raised to $0$ is $1$ .

$1 x^{8} y^{0} + 8 {(- x)}^{7} {(5 y)}^{1} + 28 {(- x)}^{6} {(5 y)}^{2} + 56 {(- x)}^{5} {(5 y)}^{3} + 70 {(- x)}^{4} {(5 y)}^{4} + 56 {(- x)}^{3} {(5 y)}^{5} + 28 {(- x)}^{2} {(5 y)}^{6} + 8 {(- x)}^{1} {(5 y)}^{7} + 1 {(- x)}^{0} {(5 y)}^{8}$

Step 4.8

Multiply $x^{8}$ by $1$ .

$x^{8} y^{0} + 8 {(- x)}^{7} {(5 y)}^{1} + 28 {(- x)}^{6} {(5 y)}^{2} + 56 {(- x)}^{5} {(5 y)}^{3} + 70 {(- x)}^{4} {(5 y)}^{4} + 56 {(- x)}^{3} {(5 y)}^{5} + 28 {(- x)}^{2} {(5 y)}^{6} + 8 {(- x)}^{1} {(5 y)}^{7} + 1 {(- x)}^{0} {(5 y)}^{8}$

Step 4.9

Anything raised to $0$ is $1$ .

$x^{8} \cdot 1 + 8 {(- x)}^{7} {(5 y)}^{1} + 28 {(- x)}^{6} {(5 y)}^{2} + 56 {(- x)}^{5} {(5 y)}^{3} + 70 {(- x)}^{4} {(5 y)}^{4} + 56 {(- x)}^{3} {(5 y)}^{5} + 28 {(- x)}^{2} {(5 y)}^{6} + 8 {(- x)}^{1} {(5 y)}^{7} + 1 {(- x)}^{0} {(5 y)}^{8}$

Step 4.10

Multiply $x^{8}$ by $1$ .

$x^{8} + 8 {(- x)}^{7} {(5 y)}^{1} + 28 {(- x)}^{6} {(5 y)}^{2} + 56 {(- x)}^{5} {(5 y)}^{3} + 70 {(- x)}^{4} {(5 y)}^{4} + 56 {(- x)}^{3} {(5 y)}^{5} + 28 {(- x)}^{2} {(5 y)}^{6} + 8 {(- x)}^{1} {(5 y)}^{7} + 1 {(- x)}^{0} {(5 y)}^{8}$

Step 4.11

Apply the product rule to $- x$ .

$x^{8} + 8 ({(- 1)}^{7} x^{7}) {(5 y)}^{1} + 28 {(- x)}^{6} {(5 y)}^{2} + 56 {(- x)}^{5} {(5 y)}^{3} + 70 {(- x)}^{4} {(5 y)}^{4} + 56 {(- x)}^{3} {(5 y)}^{5} + 28 {(- x)}^{2} {(5 y)}^{6} + 8 {(- x)}^{1} {(5 y)}^{7} + 1 {(- x)}^{0} {(5 y)}^{8}$

Step 4.12

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

$x^{8} + 8 (- x^{7}) {(5 y)}^{1} + 28 {(- x)}^{6} {(5 y)}^{2} + 56 {(- x)}^{5} {(5 y)}^{3} + 70 {(- x)}^{4} {(5 y)}^{4} + 56 {(- x)}^{3} {(5 y)}^{5} + 28 {(- x)}^{2} {(5 y)}^{6} + 8 {(- x)}^{1} {(5 y)}^{7} + 1 {(- x)}^{0} {(5 y)}^{8}$

Step 4.13

Multiply $- 1$ by $8$ .

$x^{8} - 8 x^{7} {(5 y)}^{1} + 28 {(- x)}^{6} {(5 y)}^{2} + 56 {(- x)}^{5} {(5 y)}^{3} + 70 {(- x)}^{4} {(5 y)}^{4} + 56 {(- x)}^{3} {(5 y)}^{5} + 28 {(- x)}^{2} {(5 y)}^{6} + 8 {(- x)}^{1} {(5 y)}^{7} + 1 {(- x)}^{0} {(5 y)}^{8}$

Step 4.14

Simplify.

$x^{8} - 8 x^{7} (5 y) + 28 {(- x)}^{6} {(5 y)}^{2} + 56 {(- x)}^{5} {(5 y)}^{3} + 70 {(- x)}^{4} {(5 y)}^{4} + 56 {(- x)}^{3} {(5 y)}^{5} + 28 {(- x)}^{2} {(5 y)}^{6} + 8 {(- x)}^{1} {(5 y)}^{7} + 1 {(- x)}^{0} {(5 y)}^{8}$

Step 4.15

Rewrite using the commutative property of multiplication.

$x^{8} - 8 \cdot 5 x^{7} y + 28 {(- x)}^{6} {(5 y)}^{2} + 56 {(- x)}^{5} {(5 y)}^{3} + 70 {(- x)}^{4} {(5 y)}^{4} + 56 {(- x)}^{3} {(5 y)}^{5} + 28 {(- x)}^{2} {(5 y)}^{6} + 8 {(- x)}^{1} {(5 y)}^{7} + 1 {(- x)}^{0} {(5 y)}^{8}$

Step 4.16

Multiply $- 8$ by $5$ .

$x^{8} - 40 x^{7} y + 28 {(- x)}^{6} {(5 y)}^{2} + 56 {(- x)}^{5} {(5 y)}^{3} + 70 {(- x)}^{4} {(5 y)}^{4} + 56 {(- x)}^{3} {(5 y)}^{5} + 28 {(- x)}^{2} {(5 y)}^{6} + 8 {(- x)}^{1} {(5 y)}^{7} + 1 {(- x)}^{0} {(5 y)}^{8}$

Step 4.17

Apply the product rule to $- x$ .

$x^{8} - 40 x^{7} y + 28 ({(- 1)}^{6} x^{6}) {(5 y)}^{2} + 56 {(- x)}^{5} {(5 y)}^{3} + 70 {(- x)}^{4} {(5 y)}^{4} + 56 {(- x)}^{3} {(5 y)}^{5} + 28 {(- x)}^{2} {(5 y)}^{6} + 8 {(- x)}^{1} {(5 y)}^{7} + 1 {(- x)}^{0} {(5 y)}^{8}$

Step 4.18

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

$x^{8} - 40 x^{7} y + 28 (1 x^{6}) {(5 y)}^{2} + 56 {(- x)}^{5} {(5 y)}^{3} + 70 {(- x)}^{4} {(5 y)}^{4} + 56 {(- x)}^{3} {(5 y)}^{5} + 28 {(- x)}^{2} {(5 y)}^{6} + 8 {(- x)}^{1} {(5 y)}^{7} + 1 {(- x)}^{0} {(5 y)}^{8}$

Step 4.19

Multiply $x^{6}$ by $1$ .

$x^{8} - 40 x^{7} y + 28 x^{6} {(5 y)}^{2} + 56 {(- x)}^{5} {(5 y)}^{3} + 70 {(- x)}^{4} {(5 y)}^{4} + 56 {(- x)}^{3} {(5 y)}^{5} + 28 {(- x)}^{2} {(5 y)}^{6} + 8 {(- x)}^{1} {(5 y)}^{7} + 1 {(- x)}^{0} {(5 y)}^{8}$

Step 4.20

Apply the product rule to $5 y$ .

$x^{8} - 40 x^{7} y + 28 x^{6} (5^{2} y^{2}) + 56 {(- x)}^{5} {(5 y)}^{3} + 70 {(- x)}^{4} {(5 y)}^{4} + 56 {(- x)}^{3} {(5 y)}^{5} + 28 {(- x)}^{2} {(5 y)}^{6} + 8 {(- x)}^{1} {(5 y)}^{7} + 1 {(- x)}^{0} {(5 y)}^{8}$

Step 4.21

Rewrite using the commutative property of multiplication.

$x^{8} - 40 x^{7} y + 28 \cdot 5^{2} x^{6} y^{2} + 56 {(- x)}^{5} {(5 y)}^{3} + 70 {(- x)}^{4} {(5 y)}^{4} + 56 {(- x)}^{3} {(5 y)}^{5} + 28 {(- x)}^{2} {(5 y)}^{6} + 8 {(- x)}^{1} {(5 y)}^{7} + 1 {(- x)}^{0} {(5 y)}^{8}$

Step 4.22

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

$x^{8} - 40 x^{7} y + 28 \cdot 25 x^{6} y^{2} + 56 {(- x)}^{5} {(5 y)}^{3} + 70 {(- x)}^{4} {(5 y)}^{4} + 56 {(- x)}^{3} {(5 y)}^{5} + 28 {(- x)}^{2} {(5 y)}^{6} + 8 {(- x)}^{1} {(5 y)}^{7} + 1 {(- x)}^{0} {(5 y)}^{8}$

Step 4.23

Multiply $28$ by $25$ .

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} + 56 {(- x)}^{5} {(5 y)}^{3} + 70 {(- x)}^{4} {(5 y)}^{4} + 56 {(- x)}^{3} {(5 y)}^{5} + 28 {(- x)}^{2} {(5 y)}^{6} + 8 {(- x)}^{1} {(5 y)}^{7} + 1 {(- x)}^{0} {(5 y)}^{8}$

Step 4.24

Apply the product rule to $- x$ .

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} + 56 ({(- 1)}^{5} x^{5}) {(5 y)}^{3} + 70 {(- x)}^{4} {(5 y)}^{4} + 56 {(- x)}^{3} {(5 y)}^{5} + 28 {(- x)}^{2} {(5 y)}^{6} + 8 {(- x)}^{1} {(5 y)}^{7} + 1 {(- x)}^{0} {(5 y)}^{8}$

Step 4.25

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

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} + 56 (- x^{5}) {(5 y)}^{3} + 70 {(- x)}^{4} {(5 y)}^{4} + 56 {(- x)}^{3} {(5 y)}^{5} + 28 {(- x)}^{2} {(5 y)}^{6} + 8 {(- x)}^{1} {(5 y)}^{7} + 1 {(- x)}^{0} {(5 y)}^{8}$

Step 4.26

Multiply $- 1$ by $56$ .

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} - 56 x^{5} {(5 y)}^{3} + 70 {(- x)}^{4} {(5 y)}^{4} + 56 {(- x)}^{3} {(5 y)}^{5} + 28 {(- x)}^{2} {(5 y)}^{6} + 8 {(- x)}^{1} {(5 y)}^{7} + 1 {(- x)}^{0} {(5 y)}^{8}$

Step 4.27

Apply the product rule to $5 y$ .

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} - 56 x^{5} (5^{3} y^{3}) + 70 {(- x)}^{4} {(5 y)}^{4} + 56 {(- x)}^{3} {(5 y)}^{5} + 28 {(- x)}^{2} {(5 y)}^{6} + 8 {(- x)}^{1} {(5 y)}^{7} + 1 {(- x)}^{0} {(5 y)}^{8}$

Step 4.28

Rewrite using the commutative property of multiplication.

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} - 56 \cdot 5^{3} x^{5} y^{3} + 70 {(- x)}^{4} {(5 y)}^{4} + 56 {(- x)}^{3} {(5 y)}^{5} + 28 {(- x)}^{2} {(5 y)}^{6} + 8 {(- x)}^{1} {(5 y)}^{7} + 1 {(- x)}^{0} {(5 y)}^{8}$

Step 4.29

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

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} - 56 \cdot 125 x^{5} y^{3} + 70 {(- x)}^{4} {(5 y)}^{4} + 56 {(- x)}^{3} {(5 y)}^{5} + 28 {(- x)}^{2} {(5 y)}^{6} + 8 {(- x)}^{1} {(5 y)}^{7} + 1 {(- x)}^{0} {(5 y)}^{8}$

Step 4.30

Multiply $- 56$ by $125$ .

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} - 7000 x^{5} y^{3} + 70 {(- x)}^{4} {(5 y)}^{4} + 56 {(- x)}^{3} {(5 y)}^{5} + 28 {(- x)}^{2} {(5 y)}^{6} + 8 {(- x)}^{1} {(5 y)}^{7} + 1 {(- x)}^{0} {(5 y)}^{8}$

Step 4.31

Apply the product rule to $- x$ .

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} - 7000 x^{5} y^{3} + 70 ({(- 1)}^{4} x^{4}) {(5 y)}^{4} + 56 {(- x)}^{3} {(5 y)}^{5} + 28 {(- x)}^{2} {(5 y)}^{6} + 8 {(- x)}^{1} {(5 y)}^{7} + 1 {(- x)}^{0} {(5 y)}^{8}$

Step 4.32

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

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} - 7000 x^{5} y^{3} + 70 (1 x^{4}) {(5 y)}^{4} + 56 {(- x)}^{3} {(5 y)}^{5} + 28 {(- x)}^{2} {(5 y)}^{6} + 8 {(- x)}^{1} {(5 y)}^{7} + 1 {(- x)}^{0} {(5 y)}^{8}$

Step 4.33

Multiply $x^{4}$ by $1$ .

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} - 7000 x^{5} y^{3} + 70 x^{4} {(5 y)}^{4} + 56 {(- x)}^{3} {(5 y)}^{5} + 28 {(- x)}^{2} {(5 y)}^{6} + 8 {(- x)}^{1} {(5 y)}^{7} + 1 {(- x)}^{0} {(5 y)}^{8}$

Step 4.34

Apply the product rule to $5 y$ .

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} - 7000 x^{5} y^{3} + 70 x^{4} (5^{4} y^{4}) + 56 {(- x)}^{3} {(5 y)}^{5} + 28 {(- x)}^{2} {(5 y)}^{6} + 8 {(- x)}^{1} {(5 y)}^{7} + 1 {(- x)}^{0} {(5 y)}^{8}$

Step 4.35

Rewrite using the commutative property of multiplication.

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} - 7000 x^{5} y^{3} + 70 \cdot 5^{4} x^{4} y^{4} + 56 {(- x)}^{3} {(5 y)}^{5} + 28 {(- x)}^{2} {(5 y)}^{6} + 8 {(- x)}^{1} {(5 y)}^{7} + 1 {(- x)}^{0} {(5 y)}^{8}$

Step 4.36

Raise $5$ to the power of $4$ .

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} - 7000 x^{5} y^{3} + 70 \cdot 625 x^{4} y^{4} + 56 {(- x)}^{3} {(5 y)}^{5} + 28 {(- x)}^{2} {(5 y)}^{6} + 8 {(- x)}^{1} {(5 y)}^{7} + 1 {(- x)}^{0} {(5 y)}^{8}$

Step 4.37

Multiply $70$ by $625$ .

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} - 7000 x^{5} y^{3} + 43750 x^{4} y^{4} + 56 {(- x)}^{3} {(5 y)}^{5} + 28 {(- x)}^{2} {(5 y)}^{6} + 8 {(- x)}^{1} {(5 y)}^{7} + 1 {(- x)}^{0} {(5 y)}^{8}$

Step 4.38

Apply the product rule to $- x$ .

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} - 7000 x^{5} y^{3} + 43750 x^{4} y^{4} + 56 ({(- 1)}^{3} x^{3}) {(5 y)}^{5} + 28 {(- x)}^{2} {(5 y)}^{6} + 8 {(- x)}^{1} {(5 y)}^{7} + 1 {(- x)}^{0} {(5 y)}^{8}$

Step 4.39

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

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} - 7000 x^{5} y^{3} + 43750 x^{4} y^{4} + 56 (- x^{3}) {(5 y)}^{5} + 28 {(- x)}^{2} {(5 y)}^{6} + 8 {(- x)}^{1} {(5 y)}^{7} + 1 {(- x)}^{0} {(5 y)}^{8}$

Step 4.40

Multiply $- 1$ by $56$ .

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} - 7000 x^{5} y^{3} + 43750 x^{4} y^{4} - 56 x^{3} {(5 y)}^{5} + 28 {(- x)}^{2} {(5 y)}^{6} + 8 {(- x)}^{1} {(5 y)}^{7} + 1 {(- x)}^{0} {(5 y)}^{8}$

Step 4.41

Apply the product rule to $5 y$ .

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} - 7000 x^{5} y^{3} + 43750 x^{4} y^{4} - 56 x^{3} (5^{5} y^{5}) + 28 {(- x)}^{2} {(5 y)}^{6} + 8 {(- x)}^{1} {(5 y)}^{7} + 1 {(- x)}^{0} {(5 y)}^{8}$

Step 4.42

Rewrite using the commutative property of multiplication.

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} - 7000 x^{5} y^{3} + 43750 x^{4} y^{4} - 56 \cdot 5^{5} x^{3} y^{5} + 28 {(- x)}^{2} {(5 y)}^{6} + 8 {(- x)}^{1} {(5 y)}^{7} + 1 {(- x)}^{0} {(5 y)}^{8}$

Step 4.43

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

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} - 7000 x^{5} y^{3} + 43750 x^{4} y^{4} - 56 \cdot 3125 x^{3} y^{5} + 28 {(- x)}^{2} {(5 y)}^{6} + 8 {(- x)}^{1} {(5 y)}^{7} + 1 {(- x)}^{0} {(5 y)}^{8}$

Step 4.44

Multiply $- 56$ by $3125$ .

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} - 7000 x^{5} y^{3} + 43750 x^{4} y^{4} - 175000 x^{3} y^{5} + 28 {(- x)}^{2} {(5 y)}^{6} + 8 {(- x)}^{1} {(5 y)}^{7} + 1 {(- x)}^{0} {(5 y)}^{8}$

Step 4.45

Apply the product rule to $- x$ .

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} - 7000 x^{5} y^{3} + 43750 x^{4} y^{4} - 175000 x^{3} y^{5} + 28 ({(- 1)}^{2} x^{2}) {(5 y)}^{6} + 8 {(- x)}^{1} {(5 y)}^{7} + 1 {(- x)}^{0} {(5 y)}^{8}$

Step 4.46

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

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} - 7000 x^{5} y^{3} + 43750 x^{4} y^{4} - 175000 x^{3} y^{5} + 28 (1 x^{2}) {(5 y)}^{6} + 8 {(- x)}^{1} {(5 y)}^{7} + 1 {(- x)}^{0} {(5 y)}^{8}$

Step 4.47

Multiply $x^{2}$ by $1$ .

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} - 7000 x^{5} y^{3} + 43750 x^{4} y^{4} - 175000 x^{3} y^{5} + 28 x^{2} {(5 y)}^{6} + 8 {(- x)}^{1} {(5 y)}^{7} + 1 {(- x)}^{0} {(5 y)}^{8}$

Step 4.48

Apply the product rule to $5 y$ .

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} - 7000 x^{5} y^{3} + 43750 x^{4} y^{4} - 175000 x^{3} y^{5} + 28 x^{2} (5^{6} y^{6}) + 8 {(- x)}^{1} {(5 y)}^{7} + 1 {(- x)}^{0} {(5 y)}^{8}$

Step 4.49

Rewrite using the commutative property of multiplication.

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} - 7000 x^{5} y^{3} + 43750 x^{4} y^{4} - 175000 x^{3} y^{5} + 28 \cdot 5^{6} x^{2} y^{6} + 8 {(- x)}^{1} {(5 y)}^{7} + 1 {(- x)}^{0} {(5 y)}^{8}$

Step 4.50

Raise $5$ to the power of $6$ .

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} - 7000 x^{5} y^{3} + 43750 x^{4} y^{4} - 175000 x^{3} y^{5} + 28 \cdot 15625 x^{2} y^{6} + 8 {(- x)}^{1} {(5 y)}^{7} + 1 {(- x)}^{0} {(5 y)}^{8}$

Step 4.51

Multiply $28$ by $15625$ .

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} - 7000 x^{5} y^{3} + 43750 x^{4} y^{4} - 175000 x^{3} y^{5} + 437500 x^{2} y^{6} + 8 {(- x)}^{1} {(5 y)}^{7} + 1 {(- x)}^{0} {(5 y)}^{8}$

Step 4.52

Simplify.

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} - 7000 x^{5} y^{3} + 43750 x^{4} y^{4} - 175000 x^{3} y^{5} + 437500 x^{2} y^{6} + 8 (- x) {(5 y)}^{7} + 1 {(- x)}^{0} {(5 y)}^{8}$

Step 4.53

Multiply $- 1$ by $8$ .

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} - 7000 x^{5} y^{3} + 43750 x^{4} y^{4} - 175000 x^{3} y^{5} + 437500 x^{2} y^{6} - 8 x {(5 y)}^{7} + 1 {(- x)}^{0} {(5 y)}^{8}$

Step 4.54

Apply the product rule to $5 y$ .

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} - 7000 x^{5} y^{3} + 43750 x^{4} y^{4} - 175000 x^{3} y^{5} + 437500 x^{2} y^{6} - 8 x (5^{7} y^{7}) + 1 {(- x)}^{0} {(5 y)}^{8}$

Step 4.55

Rewrite using the commutative property of multiplication.

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} - 7000 x^{5} y^{3} + 43750 x^{4} y^{4} - 175000 x^{3} y^{5} + 437500 x^{2} y^{6} - 8 \cdot 5^{7} x y^{7} + 1 {(- x)}^{0} {(5 y)}^{8}$

Step 4.56

Raise $5$ to the power of $7$ .

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} - 7000 x^{5} y^{3} + 43750 x^{4} y^{4} - 175000 x^{3} y^{5} + 437500 x^{2} y^{6} - 8 \cdot 78125 x y^{7} + 1 {(- x)}^{0} {(5 y)}^{8}$

Step 4.57

Multiply $- 8$ by $78125$ .

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} - 7000 x^{5} y^{3} + 43750 x^{4} y^{4} - 175000 x^{3} y^{5} + 437500 x^{2} y^{6} - 625000 x y^{7} + 1 {(- x)}^{0} {(5 y)}^{8}$

Step 4.58

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

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} - 7000 x^{5} y^{3} + 43750 x^{4} y^{4} - 175000 x^{3} y^{5} + 437500 x^{2} y^{6} - 625000 x y^{7} + {(- x)}^{0} {(5 y)}^{8}$

Step 4.59

Apply the product rule to $- x$ .

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} - 7000 x^{5} y^{3} + 43750 x^{4} y^{4} - 175000 x^{3} y^{5} + 437500 x^{2} y^{6} - 625000 x y^{7} + {(- 1)}^{0} x^{0} {(5 y)}^{8}$

Step 4.60

Anything raised to $0$ is $1$ .

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} - 7000 x^{5} y^{3} + 43750 x^{4} y^{4} - 175000 x^{3} y^{5} + 437500 x^{2} y^{6} - 625000 x y^{7} + 1 x^{0} {(5 y)}^{8}$

Step 4.61

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

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} - 7000 x^{5} y^{3} + 43750 x^{4} y^{4} - 175000 x^{3} y^{5} + 437500 x^{2} y^{6} - 625000 x y^{7} + x^{0} {(5 y)}^{8}$

Step 4.62

Anything raised to $0$ is $1$ .

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} - 7000 x^{5} y^{3} + 43750 x^{4} y^{4} - 175000 x^{3} y^{5} + 437500 x^{2} y^{6} - 625000 x y^{7} + 1 {(5 y)}^{8}$

Step 4.63

Multiply ${(5 y)}^{8}$ by $1$ .

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} - 7000 x^{5} y^{3} + 43750 x^{4} y^{4} - 175000 x^{3} y^{5} + 437500 x^{2} y^{6} - 625000 x y^{7} + {(5 y)}^{8}$

Step 4.64

Apply the product rule to $5 y$ .

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} - 7000 x^{5} y^{3} + 43750 x^{4} y^{4} - 175000 x^{3} y^{5} + 437500 x^{2} y^{6} - 625000 x y^{7} + 5^{8} y^{8}$

Step 4.65

Raise $5$ to the power of $8$ .

$x^{8} - 40 x^{7} y + 700 x^{6} y^{2} - 7000 x^{5} y^{3} + 43750 x^{4} y^{4} - 175000 x^{3} y^{5} + 437500 x^{2} y^{6} - 625000 x y^{7} + 390625 y^{8}$