Algebra Examples

${(x - y)}^{5} - {(x^{2} - y^{2})}^{7}$

Step 1

Use the Binomial Theorem.

$x^{5} + 5 x^{4} (- y) + 10 x^{3} {(- y)}^{2} + 10 x^{2} {(- y)}^{3} + 5 x {(- y)}^{4} + {(- y)}^{5} - {(x^{2} - y^{2})}^{7}$

Step 2

Simplify each term.

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Step 2.1

Rewrite using the commutative property of multiplication.

$x^{5} + 5 \cdot - 1 x^{4} y + 10 x^{3} {(- y)}^{2} + 10 x^{2} {(- y)}^{3} + 5 x {(- y)}^{4} + {(- y)}^{5} - {(x^{2} - y^{2})}^{7}$

Step 2.2

Multiply $5$ by $- 1$ .

$x^{5} - 5 x^{4} y + 10 x^{3} {(- y)}^{2} + 10 x^{2} {(- y)}^{3} + 5 x {(- y)}^{4} + {(- y)}^{5} - {(x^{2} - y^{2})}^{7}$

Step 2.3

Apply the product rule to $- y$ .

$x^{5} - 5 x^{4} y + 10 x^{3} ({(- 1)}^{2} y^{2}) + 10 x^{2} {(- y)}^{3} + 5 x {(- y)}^{4} + {(- y)}^{5} - {(x^{2} - y^{2})}^{7}$

Step 2.4

Rewrite using the commutative property of multiplication.

$x^{5} - 5 x^{4} y + 10 \cdot {(- 1)}^{2} x^{3} y^{2} + 10 x^{2} {(- y)}^{3} + 5 x {(- y)}^{4} + {(- y)}^{5} - {(x^{2} - y^{2})}^{7}$

Step 2.5

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

$x^{5} - 5 x^{4} y + 10 \cdot 1 x^{3} y^{2} + 10 x^{2} {(- y)}^{3} + 5 x {(- y)}^{4} + {(- y)}^{5} - {(x^{2} - y^{2})}^{7}$

Step 2.6

Multiply $10$ by $1$ .

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} + 10 x^{2} {(- y)}^{3} + 5 x {(- y)}^{4} + {(- y)}^{5} - {(x^{2} - y^{2})}^{7}$

Step 2.7

Apply the product rule to $- y$ .

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} + 10 x^{2} ({(- 1)}^{3} y^{3}) + 5 x {(- y)}^{4} + {(- y)}^{5} - {(x^{2} - y^{2})}^{7}$

Step 2.8

Rewrite using the commutative property of multiplication.

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} + 10 \cdot {(- 1)}^{3} x^{2} y^{3} + 5 x {(- y)}^{4} + {(- y)}^{5} - {(x^{2} - y^{2})}^{7}$

Step 2.9

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

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} + 10 \cdot - 1 x^{2} y^{3} + 5 x {(- y)}^{4} + {(- y)}^{5} - {(x^{2} - y^{2})}^{7}$

Step 2.10

Multiply $10$ by $- 1$ .

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x {(- y)}^{4} + {(- y)}^{5} - {(x^{2} - y^{2})}^{7}$

Step 2.11

Apply the product rule to $- y$ .

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x ({(- 1)}^{4} y^{4}) + {(- y)}^{5} - {(x^{2} - y^{2})}^{7}$

Step 2.12

Rewrite using the commutative property of multiplication.

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 \cdot {(- 1)}^{4} x y^{4} + {(- y)}^{5} - {(x^{2} - y^{2})}^{7}$

Step 2.13

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

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 \cdot 1 x y^{4} + {(- y)}^{5} - {(x^{2} - y^{2})}^{7}$

Step 2.14

Multiply $5$ by $1$ .

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} + {(- y)}^{5} - {(x^{2} - y^{2})}^{7}$

Step 2.15

Apply the product rule to $- y$ .

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} + {(- 1)}^{5} y^{5} - {(x^{2} - y^{2})}^{7}$

Step 2.16

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

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} - y^{5} - {(x^{2} - y^{2})}^{7}$

Step 3

Use the Binomial Theorem.

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} - y^{5} - ({(x^{2})}^{7} + 7 {(x^{2})}^{6} (- y^{2}) + 21 {(x^{2})}^{5} {(- y^{2})}^{2} + 35 {(x^{2})}^{4} {(- y^{2})}^{3} + 35 {(x^{2})}^{3} {(- y^{2})}^{4} + 21 {(x^{2})}^{2} {(- y^{2})}^{5} + 7 x^{2} {(- y^{2})}^{6} + {(- y^{2})}^{7})$

Step 4

Simplify each term.

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Step 4.1

Multiply the exponents in ${(x^{2})}^{7}$ .

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Step 4.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} - y^{5} - (x^{2 \cdot 7} + 7 {(x^{2})}^{6} (- y^{2}) + 21 {(x^{2})}^{5} {(- y^{2})}^{2} + 35 {(x^{2})}^{4} {(- y^{2})}^{3} + 35 {(x^{2})}^{3} {(- y^{2})}^{4} + 21 {(x^{2})}^{2} {(- y^{2})}^{5} + 7 x^{2} {(- y^{2})}^{6} + {(- y^{2})}^{7})$

Step 4.1.2

Multiply $2$ by $7$ .

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} - y^{5} - (x^{14} + 7 {(x^{2})}^{6} (- y^{2}) + 21 {(x^{2})}^{5} {(- y^{2})}^{2} + 35 {(x^{2})}^{4} {(- y^{2})}^{3} + 35 {(x^{2})}^{3} {(- y^{2})}^{4} + 21 {(x^{2})}^{2} {(- y^{2})}^{5} + 7 x^{2} {(- y^{2})}^{6} + {(- y^{2})}^{7})$

Step 4.2

Rewrite using the commutative property of multiplication.

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} - y^{5} - (x^{14} + 7 \cdot - 1 {(x^{2})}^{6} y^{2} + 21 {(x^{2})}^{5} {(- y^{2})}^{2} + 35 {(x^{2})}^{4} {(- y^{2})}^{3} + 35 {(x^{2})}^{3} {(- y^{2})}^{4} + 21 {(x^{2})}^{2} {(- y^{2})}^{5} + 7 x^{2} {(- y^{2})}^{6} + {(- y^{2})}^{7})$

Step 4.3

Multiply $7$ by $- 1$ .

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} - y^{5} - (x^{14} - 7 {(x^{2})}^{6} y^{2} + 21 {(x^{2})}^{5} {(- y^{2})}^{2} + 35 {(x^{2})}^{4} {(- y^{2})}^{3} + 35 {(x^{2})}^{3} {(- y^{2})}^{4} + 21 {(x^{2})}^{2} {(- y^{2})}^{5} + 7 x^{2} {(- y^{2})}^{6} + {(- y^{2})}^{7})$

Step 4.4

Multiply the exponents in ${(x^{2})}^{6}$ .

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Step 4.4.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} - y^{5} - (x^{14} - 7 x^{2 \cdot 6} y^{2} + 21 {(x^{2})}^{5} {(- y^{2})}^{2} + 35 {(x^{2})}^{4} {(- y^{2})}^{3} + 35 {(x^{2})}^{3} {(- y^{2})}^{4} + 21 {(x^{2})}^{2} {(- y^{2})}^{5} + 7 x^{2} {(- y^{2})}^{6} + {(- y^{2})}^{7})$

Step 4.4.2

Multiply $2$ by $6$ .

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} - y^{5} - (x^{14} - 7 x^{12} y^{2} + 21 {(x^{2})}^{5} {(- y^{2})}^{2} + 35 {(x^{2})}^{4} {(- y^{2})}^{3} + 35 {(x^{2})}^{3} {(- y^{2})}^{4} + 21 {(x^{2})}^{2} {(- y^{2})}^{5} + 7 x^{2} {(- y^{2})}^{6} + {(- y^{2})}^{7})$

Step 4.5

Multiply the exponents in ${(x^{2})}^{5}$ .

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Step 4.5.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} - y^{5} - (x^{14} - 7 x^{12} y^{2} + 21 x^{2 \cdot 5} {(- y^{2})}^{2} + 35 {(x^{2})}^{4} {(- y^{2})}^{3} + 35 {(x^{2})}^{3} {(- y^{2})}^{4} + 21 {(x^{2})}^{2} {(- y^{2})}^{5} + 7 x^{2} {(- y^{2})}^{6} + {(- y^{2})}^{7})$

Step 4.5.2

Multiply $2$ by $5$ .

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} - y^{5} - (x^{14} - 7 x^{12} y^{2} + 21 x^{10} {(- y^{2})}^{2} + 35 {(x^{2})}^{4} {(- y^{2})}^{3} + 35 {(x^{2})}^{3} {(- y^{2})}^{4} + 21 {(x^{2})}^{2} {(- y^{2})}^{5} + 7 x^{2} {(- y^{2})}^{6} + {(- y^{2})}^{7})$

Step 4.6

Apply the product rule to $- y^{2}$ .

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} - y^{5} - (x^{14} - 7 x^{12} y^{2} + 21 x^{10} ({(- 1)}^{2} {(y^{2})}^{2}) + 35 {(x^{2})}^{4} {(- y^{2})}^{3} + 35 {(x^{2})}^{3} {(- y^{2})}^{4} + 21 {(x^{2})}^{2} {(- y^{2})}^{5} + 7 x^{2} {(- y^{2})}^{6} + {(- y^{2})}^{7})$

Step 4.7

Rewrite using the commutative property of multiplication.

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} - y^{5} - (x^{14} - 7 x^{12} y^{2} + 21 \cdot {(- 1)}^{2} x^{10} {(y^{2})}^{2} + 35 {(x^{2})}^{4} {(- y^{2})}^{3} + 35 {(x^{2})}^{3} {(- y^{2})}^{4} + 21 {(x^{2})}^{2} {(- y^{2})}^{5} + 7 x^{2} {(- y^{2})}^{6} + {(- y^{2})}^{7})$

Step 4.8

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

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} - y^{5} - (x^{14} - 7 x^{12} y^{2} + 21 \cdot 1 x^{10} {(y^{2})}^{2} + 35 {(x^{2})}^{4} {(- y^{2})}^{3} + 35 {(x^{2})}^{3} {(- y^{2})}^{4} + 21 {(x^{2})}^{2} {(- y^{2})}^{5} + 7 x^{2} {(- y^{2})}^{6} + {(- y^{2})}^{7})$

Step 4.9

Multiply $21$ by $1$ .

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} - y^{5} - (x^{14} - 7 x^{12} y^{2} + 21 x^{10} {(y^{2})}^{2} + 35 {(x^{2})}^{4} {(- y^{2})}^{3} + 35 {(x^{2})}^{3} {(- y^{2})}^{4} + 21 {(x^{2})}^{2} {(- y^{2})}^{5} + 7 x^{2} {(- y^{2})}^{6} + {(- y^{2})}^{7})$

Step 4.10

Multiply the exponents in ${(y^{2})}^{2}$ .

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Step 4.10.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} - y^{5} - (x^{14} - 7 x^{12} y^{2} + 21 x^{10} y^{2 \cdot 2} + 35 {(x^{2})}^{4} {(- y^{2})}^{3} + 35 {(x^{2})}^{3} {(- y^{2})}^{4} + 21 {(x^{2})}^{2} {(- y^{2})}^{5} + 7 x^{2} {(- y^{2})}^{6} + {(- y^{2})}^{7})$

Step 4.10.2

Multiply $2$ by $2$ .

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} - y^{5} - (x^{14} - 7 x^{12} y^{2} + 21 x^{10} y^{4} + 35 {(x^{2})}^{4} {(- y^{2})}^{3} + 35 {(x^{2})}^{3} {(- y^{2})}^{4} + 21 {(x^{2})}^{2} {(- y^{2})}^{5} + 7 x^{2} {(- y^{2})}^{6} + {(- y^{2})}^{7})$

Step 4.11

Multiply the exponents in ${(x^{2})}^{4}$ .

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Step 4.11.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} - y^{5} - (x^{14} - 7 x^{12} y^{2} + 21 x^{10} y^{4} + 35 x^{2 \cdot 4} {(- y^{2})}^{3} + 35 {(x^{2})}^{3} {(- y^{2})}^{4} + 21 {(x^{2})}^{2} {(- y^{2})}^{5} + 7 x^{2} {(- y^{2})}^{6} + {(- y^{2})}^{7})$

Step 4.11.2

Multiply $2$ by $4$ .

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} - y^{5} - (x^{14} - 7 x^{12} y^{2} + 21 x^{10} y^{4} + 35 x^{8} {(- y^{2})}^{3} + 35 {(x^{2})}^{3} {(- y^{2})}^{4} + 21 {(x^{2})}^{2} {(- y^{2})}^{5} + 7 x^{2} {(- y^{2})}^{6} + {(- y^{2})}^{7})$

Step 4.12

Apply the product rule to $- y^{2}$ .

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} - y^{5} - (x^{14} - 7 x^{12} y^{2} + 21 x^{10} y^{4} + 35 x^{8} ({(- 1)}^{3} {(y^{2})}^{3}) + 35 {(x^{2})}^{3} {(- y^{2})}^{4} + 21 {(x^{2})}^{2} {(- y^{2})}^{5} + 7 x^{2} {(- y^{2})}^{6} + {(- y^{2})}^{7})$

Step 4.13

Rewrite using the commutative property of multiplication.

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} - y^{5} - (x^{14} - 7 x^{12} y^{2} + 21 x^{10} y^{4} + 35 \cdot {(- 1)}^{3} x^{8} {(y^{2})}^{3} + 35 {(x^{2})}^{3} {(- y^{2})}^{4} + 21 {(x^{2})}^{2} {(- y^{2})}^{5} + 7 x^{2} {(- y^{2})}^{6} + {(- y^{2})}^{7})$

Step 4.14

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

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} - y^{5} - (x^{14} - 7 x^{12} y^{2} + 21 x^{10} y^{4} + 35 \cdot - 1 x^{8} {(y^{2})}^{3} + 35 {(x^{2})}^{3} {(- y^{2})}^{4} + 21 {(x^{2})}^{2} {(- y^{2})}^{5} + 7 x^{2} {(- y^{2})}^{6} + {(- y^{2})}^{7})$

Step 4.15

Multiply $35$ by $- 1$ .

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} - y^{5} - (x^{14} - 7 x^{12} y^{2} + 21 x^{10} y^{4} - 35 x^{8} {(y^{2})}^{3} + 35 {(x^{2})}^{3} {(- y^{2})}^{4} + 21 {(x^{2})}^{2} {(- y^{2})}^{5} + 7 x^{2} {(- y^{2})}^{6} + {(- y^{2})}^{7})$

Step 4.16

Multiply the exponents in ${(y^{2})}^{3}$ .

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Step 4.16.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} - y^{5} - (x^{14} - 7 x^{12} y^{2} + 21 x^{10} y^{4} - 35 x^{8} y^{2 \cdot 3} + 35 {(x^{2})}^{3} {(- y^{2})}^{4} + 21 {(x^{2})}^{2} {(- y^{2})}^{5} + 7 x^{2} {(- y^{2})}^{6} + {(- y^{2})}^{7})$

Step 4.16.2

Multiply $2$ by $3$ .

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} - y^{5} - (x^{14} - 7 x^{12} y^{2} + 21 x^{10} y^{4} - 35 x^{8} y^{6} + 35 {(x^{2})}^{3} {(- y^{2})}^{4} + 21 {(x^{2})}^{2} {(- y^{2})}^{5} + 7 x^{2} {(- y^{2})}^{6} + {(- y^{2})}^{7})$

Step 4.17

Multiply the exponents in ${(x^{2})}^{3}$ .

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Step 4.17.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} - y^{5} - (x^{14} - 7 x^{12} y^{2} + 21 x^{10} y^{4} - 35 x^{8} y^{6} + 35 x^{2 \cdot 3} {(- y^{2})}^{4} + 21 {(x^{2})}^{2} {(- y^{2})}^{5} + 7 x^{2} {(- y^{2})}^{6} + {(- y^{2})}^{7})$

Step 4.17.2

Multiply $2$ by $3$ .

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} - y^{5} - (x^{14} - 7 x^{12} y^{2} + 21 x^{10} y^{4} - 35 x^{8} y^{6} + 35 x^{6} {(- y^{2})}^{4} + 21 {(x^{2})}^{2} {(- y^{2})}^{5} + 7 x^{2} {(- y^{2})}^{6} + {(- y^{2})}^{7})$

Step 4.18

Apply the product rule to $- y^{2}$ .

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} - y^{5} - (x^{14} - 7 x^{12} y^{2} + 21 x^{10} y^{4} - 35 x^{8} y^{6} + 35 x^{6} ({(- 1)}^{4} {(y^{2})}^{4}) + 21 {(x^{2})}^{2} {(- y^{2})}^{5} + 7 x^{2} {(- y^{2})}^{6} + {(- y^{2})}^{7})$

Step 4.19

Rewrite using the commutative property of multiplication.

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} - y^{5} - (x^{14} - 7 x^{12} y^{2} + 21 x^{10} y^{4} - 35 x^{8} y^{6} + 35 \cdot {(- 1)}^{4} x^{6} {(y^{2})}^{4} + 21 {(x^{2})}^{2} {(- y^{2})}^{5} + 7 x^{2} {(- y^{2})}^{6} + {(- y^{2})}^{7})$

Step 4.20

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

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} - y^{5} - (x^{14} - 7 x^{12} y^{2} + 21 x^{10} y^{4} - 35 x^{8} y^{6} + 35 \cdot 1 x^{6} {(y^{2})}^{4} + 21 {(x^{2})}^{2} {(- y^{2})}^{5} + 7 x^{2} {(- y^{2})}^{6} + {(- y^{2})}^{7})$

Step 4.21

Multiply $35$ by $1$ .

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} - y^{5} - (x^{14} - 7 x^{12} y^{2} + 21 x^{10} y^{4} - 35 x^{8} y^{6} + 35 x^{6} {(y^{2})}^{4} + 21 {(x^{2})}^{2} {(- y^{2})}^{5} + 7 x^{2} {(- y^{2})}^{6} + {(- y^{2})}^{7})$

Step 4.22

Multiply the exponents in ${(y^{2})}^{4}$ .

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Step 4.22.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} - y^{5} - (x^{14} - 7 x^{12} y^{2} + 21 x^{10} y^{4} - 35 x^{8} y^{6} + 35 x^{6} y^{2 \cdot 4} + 21 {(x^{2})}^{2} {(- y^{2})}^{5} + 7 x^{2} {(- y^{2})}^{6} + {(- y^{2})}^{7})$

Step 4.22.2

Multiply $2$ by $4$ .

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} - y^{5} - (x^{14} - 7 x^{12} y^{2} + 21 x^{10} y^{4} - 35 x^{8} y^{6} + 35 x^{6} y^{8} + 21 {(x^{2})}^{2} {(- y^{2})}^{5} + 7 x^{2} {(- y^{2})}^{6} + {(- y^{2})}^{7})$

Step 4.23

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 4.23.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} - y^{5} - (x^{14} - 7 x^{12} y^{2} + 21 x^{10} y^{4} - 35 x^{8} y^{6} + 35 x^{6} y^{8} + 21 x^{2 \cdot 2} {(- y^{2})}^{5} + 7 x^{2} {(- y^{2})}^{6} + {(- y^{2})}^{7})$

Step 4.23.2

Multiply $2$ by $2$ .

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} - y^{5} - (x^{14} - 7 x^{12} y^{2} + 21 x^{10} y^{4} - 35 x^{8} y^{6} + 35 x^{6} y^{8} + 21 x^{4} {(- y^{2})}^{5} + 7 x^{2} {(- y^{2})}^{6} + {(- y^{2})}^{7})$

Step 4.24

Apply the product rule to $- y^{2}$ .

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} - y^{5} - (x^{14} - 7 x^{12} y^{2} + 21 x^{10} y^{4} - 35 x^{8} y^{6} + 35 x^{6} y^{8} + 21 x^{4} ({(- 1)}^{5} {(y^{2})}^{5}) + 7 x^{2} {(- y^{2})}^{6} + {(- y^{2})}^{7})$

Step 4.25

Rewrite using the commutative property of multiplication.

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} - y^{5} - (x^{14} - 7 x^{12} y^{2} + 21 x^{10} y^{4} - 35 x^{8} y^{6} + 35 x^{6} y^{8} + 21 \cdot {(- 1)}^{5} x^{4} {(y^{2})}^{5} + 7 x^{2} {(- y^{2})}^{6} + {(- y^{2})}^{7})$

Step 4.26

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

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} - y^{5} - (x^{14} - 7 x^{12} y^{2} + 21 x^{10} y^{4} - 35 x^{8} y^{6} + 35 x^{6} y^{8} + 21 \cdot - 1 x^{4} {(y^{2})}^{5} + 7 x^{2} {(- y^{2})}^{6} + {(- y^{2})}^{7})$

Step 4.27

Multiply $21$ by $- 1$ .

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} - y^{5} - (x^{14} - 7 x^{12} y^{2} + 21 x^{10} y^{4} - 35 x^{8} y^{6} + 35 x^{6} y^{8} - 21 x^{4} {(y^{2})}^{5} + 7 x^{2} {(- y^{2})}^{6} + {(- y^{2})}^{7})$

Step 4.28

Multiply the exponents in ${(y^{2})}^{5}$ .

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Step 4.28.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} - y^{5} - (x^{14} - 7 x^{12} y^{2} + 21 x^{10} y^{4} - 35 x^{8} y^{6} + 35 x^{6} y^{8} - 21 x^{4} y^{2 \cdot 5} + 7 x^{2} {(- y^{2})}^{6} + {(- y^{2})}^{7})$

Step 4.28.2

Multiply $2$ by $5$ .

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} - y^{5} - (x^{14} - 7 x^{12} y^{2} + 21 x^{10} y^{4} - 35 x^{8} y^{6} + 35 x^{6} y^{8} - 21 x^{4} y^{10} + 7 x^{2} {(- y^{2})}^{6} + {(- y^{2})}^{7})$

Step 4.29

Apply the product rule to $- y^{2}$ .

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} - y^{5} - (x^{14} - 7 x^{12} y^{2} + 21 x^{10} y^{4} - 35 x^{8} y^{6} + 35 x^{6} y^{8} - 21 x^{4} y^{10} + 7 x^{2} ({(- 1)}^{6} {(y^{2})}^{6}) + {(- y^{2})}^{7})$

Step 4.30

Rewrite using the commutative property of multiplication.

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} - y^{5} - (x^{14} - 7 x^{12} y^{2} + 21 x^{10} y^{4} - 35 x^{8} y^{6} + 35 x^{6} y^{8} - 21 x^{4} y^{10} + 7 \cdot {(- 1)}^{6} x^{2} {(y^{2})}^{6} + {(- y^{2})}^{7})$

Step 4.31

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

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} - y^{5} - (x^{14} - 7 x^{12} y^{2} + 21 x^{10} y^{4} - 35 x^{8} y^{6} + 35 x^{6} y^{8} - 21 x^{4} y^{10} + 7 \cdot 1 x^{2} {(y^{2})}^{6} + {(- y^{2})}^{7})$

Step 4.32

Multiply $7$ by $1$ .

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} - y^{5} - (x^{14} - 7 x^{12} y^{2} + 21 x^{10} y^{4} - 35 x^{8} y^{6} + 35 x^{6} y^{8} - 21 x^{4} y^{10} + 7 x^{2} {(y^{2})}^{6} + {(- y^{2})}^{7})$

Step 4.33

Multiply the exponents in ${(y^{2})}^{6}$ .

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Step 4.33.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} - y^{5} - (x^{14} - 7 x^{12} y^{2} + 21 x^{10} y^{4} - 35 x^{8} y^{6} + 35 x^{6} y^{8} - 21 x^{4} y^{10} + 7 x^{2} y^{2 \cdot 6} + {(- y^{2})}^{7})$

Step 4.33.2

Multiply $2$ by $6$ .

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} - y^{5} - (x^{14} - 7 x^{12} y^{2} + 21 x^{10} y^{4} - 35 x^{8} y^{6} + 35 x^{6} y^{8} - 21 x^{4} y^{10} + 7 x^{2} y^{12} + {(- y^{2})}^{7})$

Step 4.34

Apply the product rule to $- y^{2}$ .

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} - y^{5} - (x^{14} - 7 x^{12} y^{2} + 21 x^{10} y^{4} - 35 x^{8} y^{6} + 35 x^{6} y^{8} - 21 x^{4} y^{10} + 7 x^{2} y^{12} + {(- 1)}^{7} {(y^{2})}^{7})$

Step 4.35

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

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} - y^{5} - (x^{14} - 7 x^{12} y^{2} + 21 x^{10} y^{4} - 35 x^{8} y^{6} + 35 x^{6} y^{8} - 21 x^{4} y^{10} + 7 x^{2} y^{12} - {(y^{2})}^{7})$

Step 4.36

Multiply the exponents in ${(y^{2})}^{7}$ .

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Step 4.36.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} - y^{5} - (x^{14} - 7 x^{12} y^{2} + 21 x^{10} y^{4} - 35 x^{8} y^{6} + 35 x^{6} y^{8} - 21 x^{4} y^{10} + 7 x^{2} y^{12} - y^{2 \cdot 7})$

Step 4.36.2

Multiply $2$ by $7$ .

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} - y^{5} - (x^{14} - 7 x^{12} y^{2} + 21 x^{10} y^{4} - 35 x^{8} y^{6} + 35 x^{6} y^{8} - 21 x^{4} y^{10} + 7 x^{2} y^{12} - y^{14})$

Step 5

Apply the distributive property.

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} - y^{5} - x^{14} - (- 7 x^{12} y^{2}) - (21 x^{10} y^{4}) - (- 35 x^{8} y^{6}) - (35 x^{6} y^{8}) - (- 21 x^{4} y^{10}) - (7 x^{2} y^{12}) - - y^{14}$

Step 6

Simplify.

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Step 6.1

Multiply $- 7$ by $- 1$ .

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} - y^{5} - x^{14} + 7 (x^{12} y^{2}) - (21 x^{10} y^{4}) - (- 35 x^{8} y^{6}) - (35 x^{6} y^{8}) - (- 21 x^{4} y^{10}) - (7 x^{2} y^{12}) - - y^{14}$

Step 6.2

Multiply $21$ by $- 1$ .

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} - y^{5} - x^{14} + 7 (x^{12} y^{2}) - 21 (x^{10} y^{4}) - (- 35 x^{8} y^{6}) - (35 x^{6} y^{8}) - (- 21 x^{4} y^{10}) - (7 x^{2} y^{12}) - - y^{14}$

Step 6.3

Multiply $- 35$ by $- 1$ .

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} - y^{5} - x^{14} + 7 (x^{12} y^{2}) - 21 (x^{10} y^{4}) + 35 (x^{8} y^{6}) - (35 x^{6} y^{8}) - (- 21 x^{4} y^{10}) - (7 x^{2} y^{12}) - - y^{14}$

Step 6.4

Multiply $35$ by $- 1$ .

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} - y^{5} - x^{14} + 7 (x^{12} y^{2}) - 21 (x^{10} y^{4}) + 35 (x^{8} y^{6}) - 35 (x^{6} y^{8}) - (- 21 x^{4} y^{10}) - (7 x^{2} y^{12}) - - y^{14}$

Step 6.5

Multiply $- 21$ by $- 1$ .

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} - y^{5} - x^{14} + 7 (x^{12} y^{2}) - 21 (x^{10} y^{4}) + 35 (x^{8} y^{6}) - 35 (x^{6} y^{8}) + 21 (x^{4} y^{10}) - (7 x^{2} y^{12}) - - y^{14}$

Step 6.6

Multiply $7$ by $- 1$ .

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} - y^{5} - x^{14} + 7 (x^{12} y^{2}) - 21 (x^{10} y^{4}) + 35 (x^{8} y^{6}) - 35 (x^{6} y^{8}) + 21 (x^{4} y^{10}) - 7 (x^{2} y^{12}) - - y^{14}$

Step 6.7

Multiply $- - y^{14}$ .

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Step 6.7.1

Multiply $- 1$ by $- 1$ .

Step 6.7.2

Multiply $y^{14}$ by $1$ .

Step 7

Simplify each term.

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Step 7.1

Remove parentheses.

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} - y^{5} - x^{14} + 7 x^{12} y^{2} - 21 (x^{10} y^{4}) + 35 (x^{8} y^{6}) - 35 (x^{6} y^{8}) + 21 (x^{4} y^{10}) - 7 (x^{2} y^{12}) + y^{14}$

Step 7.2

Remove parentheses.

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} - y^{5} - x^{14} + 7 x^{12} y^{2} - 21 x^{10} y^{4} + 35 (x^{8} y^{6}) - 35 (x^{6} y^{8}) + 21 (x^{4} y^{10}) - 7 (x^{2} y^{12}) + y^{14}$

Step 7.3

Remove parentheses.

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} - y^{5} - x^{14} + 7 x^{12} y^{2} - 21 x^{10} y^{4} + 35 x^{8} y^{6} - 35 (x^{6} y^{8}) + 21 (x^{4} y^{10}) - 7 (x^{2} y^{12}) + y^{14}$

Step 7.4

Remove parentheses.

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} - y^{5} - x^{14} + 7 x^{12} y^{2} - 21 x^{10} y^{4} + 35 x^{8} y^{6} - 35 x^{6} y^{8} + 21 (x^{4} y^{10}) - 7 (x^{2} y^{12}) + y^{14}$

Step 7.5

Remove parentheses.

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} - y^{5} - x^{14} + 7 x^{12} y^{2} - 21 x^{10} y^{4} + 35 x^{8} y^{6} - 35 x^{6} y^{8} + 21 x^{4} y^{10} - 7 (x^{2} y^{12}) + y^{14}$

Step 7.6

Remove parentheses.

$x^{5} - 5 x^{4} y + 10 x^{3} y^{2} - 10 x^{2} y^{3} + 5 x y^{4} - y^{5} - x^{14} + 7 x^{12} y^{2} - 21 x^{10} y^{4} + 35 x^{8} y^{6} - 35 x^{6} y^{8} + 21 x^{4} y^{10} - 7 x^{2} y^{12} + y^{14}$