Algebra Examples

${(2 x - y^{2})}^{6}$

Step 1

Pascal's Triangle can be displayed as such:

$1$

$1 - 1$

$1 - 2 - 1$

$1 - 3 - 3 - 1$

$1 - 4 - 6 - 4 - 1$

$1 - 5 - 10 - 10 - 5 - 1$

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

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 ${(2 x - y^{2})}^{6}$ , $n = 6$ so the coefficients of the expansion will correspond with line $7$ .

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

$1 a^{6} b^{0} + 6 a^{5} b + 15 a^{4} b^{2} + 20 a^{3} b^{3} + 15 a^{2} b^{4} + 6 a b^{5} + 1 a^{0} b^{6}$

Step 3

Substitute the actual values of $a$ $2 x$ and $b$ $- y^{2}$ into the expression.

$1 {(2 x)}^{6} {(- y^{2})}^{0} + 6 {(2 x)}^{5} {(- y^{2})}^{1} + 15 {(2 x)}^{4} {(- y^{2})}^{2} + 20 {(2 x)}^{3} {(- y^{2})}^{3} + 15 {(2 x)}^{2} {(- y^{2})}^{4} + 6 {(2 x)}^{1} {(- y^{2})}^{5} + 1 {(2 x)}^{0} {(- y^{2})}^{6}$

Step 4

Simplify each term.

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Step 4.1

Multiply ${(2 x)}^{6}$ by $1$ .

${(2 x)}^{6} {(- y^{2})}^{0} + 6 {(2 x)}^{5} {(- y^{2})}^{1} + 15 {(2 x)}^{4} {(- y^{2})}^{2} + 20 {(2 x)}^{3} {(- y^{2})}^{3} + 15 {(2 x)}^{2} {(- y^{2})}^{4} + 6 {(2 x)}^{1} {(- y^{2})}^{5} + 1 {(2 x)}^{0} {(- y^{2})}^{6}$

Step 4.2

Apply the product rule to $2 x$ .

$2^{6} x^{6} {(- y^{2})}^{0} + 6 {(2 x)}^{5} {(- y^{2})}^{1} + 15 {(2 x)}^{4} {(- y^{2})}^{2} + 20 {(2 x)}^{3} {(- y^{2})}^{3} + 15 {(2 x)}^{2} {(- y^{2})}^{4} + 6 {(2 x)}^{1} {(- y^{2})}^{5} + 1 {(2 x)}^{0} {(- y^{2})}^{6}$

Step 4.3

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

$64 x^{6} {(- y^{2})}^{0} + 6 {(2 x)}^{5} {(- y^{2})}^{1} + 15 {(2 x)}^{4} {(- y^{2})}^{2} + 20 {(2 x)}^{3} {(- y^{2})}^{3} + 15 {(2 x)}^{2} {(- y^{2})}^{4} + 6 {(2 x)}^{1} {(- y^{2})}^{5} + 1 {(2 x)}^{0} {(- y^{2})}^{6}$

Step 4.4

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

$64 x^{6} ({(- 1)}^{0} {(y^{2})}^{0}) + 6 {(2 x)}^{5} {(- y^{2})}^{1} + 15 {(2 x)}^{4} {(- y^{2})}^{2} + 20 {(2 x)}^{3} {(- y^{2})}^{3} + 15 {(2 x)}^{2} {(- y^{2})}^{4} + 6 {(2 x)}^{1} {(- y^{2})}^{5} + 1 {(2 x)}^{0} {(- y^{2})}^{6}$

Step 4.5

Rewrite using the commutative property of multiplication.

$64 \cdot {(- 1)}^{0} x^{6} {(y^{2})}^{0} + 6 {(2 x)}^{5} {(- y^{2})}^{1} + 15 {(2 x)}^{4} {(- y^{2})}^{2} + 20 {(2 x)}^{3} {(- y^{2})}^{3} + 15 {(2 x)}^{2} {(- y^{2})}^{4} + 6 {(2 x)}^{1} {(- y^{2})}^{5} + 1 {(2 x)}^{0} {(- y^{2})}^{6}$

Step 4.6

Anything raised to $0$ is $1$ .

$64 \cdot 1 x^{6} {(y^{2})}^{0} + 6 {(2 x)}^{5} {(- y^{2})}^{1} + 15 {(2 x)}^{4} {(- y^{2})}^{2} + 20 {(2 x)}^{3} {(- y^{2})}^{3} + 15 {(2 x)}^{2} {(- y^{2})}^{4} + 6 {(2 x)}^{1} {(- y^{2})}^{5} + 1 {(2 x)}^{0} {(- y^{2})}^{6}$

Step 4.7

Multiply $64$ by $1$ .

$64 x^{6} {(y^{2})}^{0} + 6 {(2 x)}^{5} {(- y^{2})}^{1} + 15 {(2 x)}^{4} {(- y^{2})}^{2} + 20 {(2 x)}^{3} {(- y^{2})}^{3} + 15 {(2 x)}^{2} {(- y^{2})}^{4} + 6 {(2 x)}^{1} {(- y^{2})}^{5} + 1 {(2 x)}^{0} {(- y^{2})}^{6}$

Step 4.8

Multiply the exponents in ${(y^{2})}^{0}$ .

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Step 4.8.1

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

$64 x^{6} y^{2 \cdot 0} + 6 {(2 x)}^{5} {(- y^{2})}^{1} + 15 {(2 x)}^{4} {(- y^{2})}^{2} + 20 {(2 x)}^{3} {(- y^{2})}^{3} + 15 {(2 x)}^{2} {(- y^{2})}^{4} + 6 {(2 x)}^{1} {(- y^{2})}^{5} + 1 {(2 x)}^{0} {(- y^{2})}^{6}$

Step 4.8.2

Multiply $2$ by $0$ .

$64 x^{6} y^{0} + 6 {(2 x)}^{5} {(- y^{2})}^{1} + 15 {(2 x)}^{4} {(- y^{2})}^{2} + 20 {(2 x)}^{3} {(- y^{2})}^{3} + 15 {(2 x)}^{2} {(- y^{2})}^{4} + 6 {(2 x)}^{1} {(- y^{2})}^{5} + 1 {(2 x)}^{0} {(- y^{2})}^{6}$

Step 4.9

Anything raised to $0$ is $1$ .

$64 x^{6} \cdot 1 + 6 {(2 x)}^{5} {(- y^{2})}^{1} + 15 {(2 x)}^{4} {(- y^{2})}^{2} + 20 {(2 x)}^{3} {(- y^{2})}^{3} + 15 {(2 x)}^{2} {(- y^{2})}^{4} + 6 {(2 x)}^{1} {(- y^{2})}^{5} + 1 {(2 x)}^{0} {(- y^{2})}^{6}$

Step 4.10

Multiply $64$ by $1$ .

$64 x^{6} + 6 {(2 x)}^{5} {(- y^{2})}^{1} + 15 {(2 x)}^{4} {(- y^{2})}^{2} + 20 {(2 x)}^{3} {(- y^{2})}^{3} + 15 {(2 x)}^{2} {(- y^{2})}^{4} + 6 {(2 x)}^{1} {(- y^{2})}^{5} + 1 {(2 x)}^{0} {(- y^{2})}^{6}$

Step 4.11

Apply the product rule to $2 x$ .

$64 x^{6} + 6 (2^{5} x^{5}) {(- y^{2})}^{1} + 15 {(2 x)}^{4} {(- y^{2})}^{2} + 20 {(2 x)}^{3} {(- y^{2})}^{3} + 15 {(2 x)}^{2} {(- y^{2})}^{4} + 6 {(2 x)}^{1} {(- y^{2})}^{5} + 1 {(2 x)}^{0} {(- y^{2})}^{6}$

Step 4.12

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

$64 x^{6} + 6 (32 x^{5}) {(- y^{2})}^{1} + 15 {(2 x)}^{4} {(- y^{2})}^{2} + 20 {(2 x)}^{3} {(- y^{2})}^{3} + 15 {(2 x)}^{2} {(- y^{2})}^{4} + 6 {(2 x)}^{1} {(- y^{2})}^{5} + 1 {(2 x)}^{0} {(- y^{2})}^{6}$

Step 4.13

Multiply $32$ by $6$ .

$64 x^{6} + 192 x^{5} {(- y^{2})}^{1} + 15 {(2 x)}^{4} {(- y^{2})}^{2} + 20 {(2 x)}^{3} {(- y^{2})}^{3} + 15 {(2 x)}^{2} {(- y^{2})}^{4} + 6 {(2 x)}^{1} {(- y^{2})}^{5} + 1 {(2 x)}^{0} {(- y^{2})}^{6}$

Step 4.14

Simplify.

$64 x^{6} + 192 x^{5} (- y^{2}) + 15 {(2 x)}^{4} {(- y^{2})}^{2} + 20 {(2 x)}^{3} {(- y^{2})}^{3} + 15 {(2 x)}^{2} {(- y^{2})}^{4} + 6 {(2 x)}^{1} {(- y^{2})}^{5} + 1 {(2 x)}^{0} {(- y^{2})}^{6}$

Step 4.15

Rewrite using the commutative property of multiplication.

$64 x^{6} + 192 \cdot - 1 x^{5} y^{2} + 15 {(2 x)}^{4} {(- y^{2})}^{2} + 20 {(2 x)}^{3} {(- y^{2})}^{3} + 15 {(2 x)}^{2} {(- y^{2})}^{4} + 6 {(2 x)}^{1} {(- y^{2})}^{5} + 1 {(2 x)}^{0} {(- y^{2})}^{6}$

Step 4.16

Multiply $192$ by $- 1$ .

$64 x^{6} - 192 x^{5} y^{2} + 15 {(2 x)}^{4} {(- y^{2})}^{2} + 20 {(2 x)}^{3} {(- y^{2})}^{3} + 15 {(2 x)}^{2} {(- y^{2})}^{4} + 6 {(2 x)}^{1} {(- y^{2})}^{5} + 1 {(2 x)}^{0} {(- y^{2})}^{6}$

Step 4.17

Apply the product rule to $2 x$ .

$64 x^{6} - 192 x^{5} y^{2} + 15 (2^{4} x^{4}) {(- y^{2})}^{2} + 20 {(2 x)}^{3} {(- y^{2})}^{3} + 15 {(2 x)}^{2} {(- y^{2})}^{4} + 6 {(2 x)}^{1} {(- y^{2})}^{5} + 1 {(2 x)}^{0} {(- y^{2})}^{6}$

Step 4.18

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

$64 x^{6} - 192 x^{5} y^{2} + 15 (16 x^{4}) {(- y^{2})}^{2} + 20 {(2 x)}^{3} {(- y^{2})}^{3} + 15 {(2 x)}^{2} {(- y^{2})}^{4} + 6 {(2 x)}^{1} {(- y^{2})}^{5} + 1 {(2 x)}^{0} {(- y^{2})}^{6}$

Step 4.19

Multiply $16$ by $15$ .

$64 x^{6} - 192 x^{5} y^{2} + 240 x^{4} {(- y^{2})}^{2} + 20 {(2 x)}^{3} {(- y^{2})}^{3} + 15 {(2 x)}^{2} {(- y^{2})}^{4} + 6 {(2 x)}^{1} {(- y^{2})}^{5} + 1 {(2 x)}^{0} {(- y^{2})}^{6}$

Step 4.20

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

$64 x^{6} - 192 x^{5} y^{2} + 240 x^{4} ({(- 1)}^{2} {(y^{2})}^{2}) + 20 {(2 x)}^{3} {(- y^{2})}^{3} + 15 {(2 x)}^{2} {(- y^{2})}^{4} + 6 {(2 x)}^{1} {(- y^{2})}^{5} + 1 {(2 x)}^{0} {(- y^{2})}^{6}$

Step 4.21

Rewrite using the commutative property of multiplication.

$64 x^{6} - 192 x^{5} y^{2} + 240 \cdot {(- 1)}^{2} x^{4} {(y^{2})}^{2} + 20 {(2 x)}^{3} {(- y^{2})}^{3} + 15 {(2 x)}^{2} {(- y^{2})}^{4} + 6 {(2 x)}^{1} {(- y^{2})}^{5} + 1 {(2 x)}^{0} {(- y^{2})}^{6}$

Step 4.22

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

$64 x^{6} - 192 x^{5} y^{2} + 240 \cdot 1 x^{4} {(y^{2})}^{2} + 20 {(2 x)}^{3} {(- y^{2})}^{3} + 15 {(2 x)}^{2} {(- y^{2})}^{4} + 6 {(2 x)}^{1} {(- y^{2})}^{5} + 1 {(2 x)}^{0} {(- y^{2})}^{6}$

Step 4.23

Multiply $240$ by $1$ .

$64 x^{6} - 192 x^{5} y^{2} + 240 x^{4} {(y^{2})}^{2} + 20 {(2 x)}^{3} {(- y^{2})}^{3} + 15 {(2 x)}^{2} {(- y^{2})}^{4} + 6 {(2 x)}^{1} {(- y^{2})}^{5} + 1 {(2 x)}^{0} {(- y^{2})}^{6}$

Step 4.24

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

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Step 4.24.1

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

$64 x^{6} - 192 x^{5} y^{2} + 240 x^{4} y^{2 \cdot 2} + 20 {(2 x)}^{3} {(- y^{2})}^{3} + 15 {(2 x)}^{2} {(- y^{2})}^{4} + 6 {(2 x)}^{1} {(- y^{2})}^{5} + 1 {(2 x)}^{0} {(- y^{2})}^{6}$

Step 4.24.2

Multiply $2$ by $2$ .

$64 x^{6} - 192 x^{5} y^{2} + 240 x^{4} y^{4} + 20 {(2 x)}^{3} {(- y^{2})}^{3} + 15 {(2 x)}^{2} {(- y^{2})}^{4} + 6 {(2 x)}^{1} {(- y^{2})}^{5} + 1 {(2 x)}^{0} {(- y^{2})}^{6}$

Step 4.25

Apply the product rule to $2 x$ .

$64 x^{6} - 192 x^{5} y^{2} + 240 x^{4} y^{4} + 20 (2^{3} x^{3}) {(- y^{2})}^{3} + 15 {(2 x)}^{2} {(- y^{2})}^{4} + 6 {(2 x)}^{1} {(- y^{2})}^{5} + 1 {(2 x)}^{0} {(- y^{2})}^{6}$

Step 4.26

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

$64 x^{6} - 192 x^{5} y^{2} + 240 x^{4} y^{4} + 20 (8 x^{3}) {(- y^{2})}^{3} + 15 {(2 x)}^{2} {(- y^{2})}^{4} + 6 {(2 x)}^{1} {(- y^{2})}^{5} + 1 {(2 x)}^{0} {(- y^{2})}^{6}$

Step 4.27

Multiply $8$ by $20$ .

$64 x^{6} - 192 x^{5} y^{2} + 240 x^{4} y^{4} + 160 x^{3} {(- y^{2})}^{3} + 15 {(2 x)}^{2} {(- y^{2})}^{4} + 6 {(2 x)}^{1} {(- y^{2})}^{5} + 1 {(2 x)}^{0} {(- y^{2})}^{6}$

Step 4.28

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

$64 x^{6} - 192 x^{5} y^{2} + 240 x^{4} y^{4} + 160 x^{3} ({(- 1)}^{3} {(y^{2})}^{3}) + 15 {(2 x)}^{2} {(- y^{2})}^{4} + 6 {(2 x)}^{1} {(- y^{2})}^{5} + 1 {(2 x)}^{0} {(- y^{2})}^{6}$

Step 4.29

Rewrite using the commutative property of multiplication.

$64 x^{6} - 192 x^{5} y^{2} + 240 x^{4} y^{4} + 160 \cdot {(- 1)}^{3} x^{3} {(y^{2})}^{3} + 15 {(2 x)}^{2} {(- y^{2})}^{4} + 6 {(2 x)}^{1} {(- y^{2})}^{5} + 1 {(2 x)}^{0} {(- y^{2})}^{6}$

Step 4.30

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

$64 x^{6} - 192 x^{5} y^{2} + 240 x^{4} y^{4} + 160 \cdot - 1 x^{3} {(y^{2})}^{3} + 15 {(2 x)}^{2} {(- y^{2})}^{4} + 6 {(2 x)}^{1} {(- y^{2})}^{5} + 1 {(2 x)}^{0} {(- y^{2})}^{6}$

Step 4.31

Multiply $160$ by $- 1$ .

$64 x^{6} - 192 x^{5} y^{2} + 240 x^{4} y^{4} - 160 x^{3} {(y^{2})}^{3} + 15 {(2 x)}^{2} {(- y^{2})}^{4} + 6 {(2 x)}^{1} {(- y^{2})}^{5} + 1 {(2 x)}^{0} {(- y^{2})}^{6}$

Step 4.32

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

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Step 4.32.1

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

$64 x^{6} - 192 x^{5} y^{2} + 240 x^{4} y^{4} - 160 x^{3} y^{2 \cdot 3} + 15 {(2 x)}^{2} {(- y^{2})}^{4} + 6 {(2 x)}^{1} {(- y^{2})}^{5} + 1 {(2 x)}^{0} {(- y^{2})}^{6}$

Step 4.32.2

Multiply $2$ by $3$ .

$64 x^{6} - 192 x^{5} y^{2} + 240 x^{4} y^{4} - 160 x^{3} y^{6} + 15 {(2 x)}^{2} {(- y^{2})}^{4} + 6 {(2 x)}^{1} {(- y^{2})}^{5} + 1 {(2 x)}^{0} {(- y^{2})}^{6}$

Step 4.33

Apply the product rule to $2 x$ .

$64 x^{6} - 192 x^{5} y^{2} + 240 x^{4} y^{4} - 160 x^{3} y^{6} + 15 (2^{2} x^{2}) {(- y^{2})}^{4} + 6 {(2 x)}^{1} {(- y^{2})}^{5} + 1 {(2 x)}^{0} {(- y^{2})}^{6}$

Step 4.34

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

$64 x^{6} - 192 x^{5} y^{2} + 240 x^{4} y^{4} - 160 x^{3} y^{6} + 15 (4 x^{2}) {(- y^{2})}^{4} + 6 {(2 x)}^{1} {(- y^{2})}^{5} + 1 {(2 x)}^{0} {(- y^{2})}^{6}$

Step 4.35

Multiply $4$ by $15$ .

$64 x^{6} - 192 x^{5} y^{2} + 240 x^{4} y^{4} - 160 x^{3} y^{6} + 60 x^{2} {(- y^{2})}^{4} + 6 {(2 x)}^{1} {(- y^{2})}^{5} + 1 {(2 x)}^{0} {(- y^{2})}^{6}$

Step 4.36

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

$64 x^{6} - 192 x^{5} y^{2} + 240 x^{4} y^{4} - 160 x^{3} y^{6} + 60 x^{2} ({(- 1)}^{4} {(y^{2})}^{4}) + 6 {(2 x)}^{1} {(- y^{2})}^{5} + 1 {(2 x)}^{0} {(- y^{2})}^{6}$

Step 4.37

Rewrite using the commutative property of multiplication.

$64 x^{6} - 192 x^{5} y^{2} + 240 x^{4} y^{4} - 160 x^{3} y^{6} + 60 \cdot {(- 1)}^{4} x^{2} {(y^{2})}^{4} + 6 {(2 x)}^{1} {(- y^{2})}^{5} + 1 {(2 x)}^{0} {(- y^{2})}^{6}$

Step 4.38

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

$64 x^{6} - 192 x^{5} y^{2} + 240 x^{4} y^{4} - 160 x^{3} y^{6} + 60 \cdot 1 x^{2} {(y^{2})}^{4} + 6 {(2 x)}^{1} {(- y^{2})}^{5} + 1 {(2 x)}^{0} {(- y^{2})}^{6}$

Step 4.39

Multiply $60$ by $1$ .

$64 x^{6} - 192 x^{5} y^{2} + 240 x^{4} y^{4} - 160 x^{3} y^{6} + 60 x^{2} {(y^{2})}^{4} + 6 {(2 x)}^{1} {(- y^{2})}^{5} + 1 {(2 x)}^{0} {(- y^{2})}^{6}$

Step 4.40

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

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Step 4.40.1

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

$64 x^{6} - 192 x^{5} y^{2} + 240 x^{4} y^{4} - 160 x^{3} y^{6} + 60 x^{2} y^{2 \cdot 4} + 6 {(2 x)}^{1} {(- y^{2})}^{5} + 1 {(2 x)}^{0} {(- y^{2})}^{6}$

Step 4.40.2

Multiply $2$ by $4$ .

$64 x^{6} - 192 x^{5} y^{2} + 240 x^{4} y^{4} - 160 x^{3} y^{6} + 60 x^{2} y^{8} + 6 {(2 x)}^{1} {(- y^{2})}^{5} + 1 {(2 x)}^{0} {(- y^{2})}^{6}$

Step 4.41

Simplify.

$64 x^{6} - 192 x^{5} y^{2} + 240 x^{4} y^{4} - 160 x^{3} y^{6} + 60 x^{2} y^{8} + 6 (2 x) {(- y^{2})}^{5} + 1 {(2 x)}^{0} {(- y^{2})}^{6}$

Step 4.42

Multiply $2$ by $6$ .

$64 x^{6} - 192 x^{5} y^{2} + 240 x^{4} y^{4} - 160 x^{3} y^{6} + 60 x^{2} y^{8} + 12 x {(- y^{2})}^{5} + 1 {(2 x)}^{0} {(- y^{2})}^{6}$

Step 4.43

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

$64 x^{6} - 192 x^{5} y^{2} + 240 x^{4} y^{4} - 160 x^{3} y^{6} + 60 x^{2} y^{8} + 12 x ({(- 1)}^{5} {(y^{2})}^{5}) + 1 {(2 x)}^{0} {(- y^{2})}^{6}$

Step 4.44

Rewrite using the commutative property of multiplication.

$64 x^{6} - 192 x^{5} y^{2} + 240 x^{4} y^{4} - 160 x^{3} y^{6} + 60 x^{2} y^{8} + 12 \cdot {(- 1)}^{5} x {(y^{2})}^{5} + 1 {(2 x)}^{0} {(- y^{2})}^{6}$

Step 4.45

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

$64 x^{6} - 192 x^{5} y^{2} + 240 x^{4} y^{4} - 160 x^{3} y^{6} + 60 x^{2} y^{8} + 12 \cdot - 1 x {(y^{2})}^{5} + 1 {(2 x)}^{0} {(- y^{2})}^{6}$

Step 4.46

Multiply $12$ by $- 1$ .

$64 x^{6} - 192 x^{5} y^{2} + 240 x^{4} y^{4} - 160 x^{3} y^{6} + 60 x^{2} y^{8} - 12 x {(y^{2})}^{5} + 1 {(2 x)}^{0} {(- y^{2})}^{6}$

Step 4.47

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

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Step 4.47.1

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

$64 x^{6} - 192 x^{5} y^{2} + 240 x^{4} y^{4} - 160 x^{3} y^{6} + 60 x^{2} y^{8} - 12 x y^{2 \cdot 5} + 1 {(2 x)}^{0} {(- y^{2})}^{6}$

Step 4.47.2

Multiply $2$ by $5$ .

$64 x^{6} - 192 x^{5} y^{2} + 240 x^{4} y^{4} - 160 x^{3} y^{6} + 60 x^{2} y^{8} - 12 x y^{10} + 1 {(2 x)}^{0} {(- y^{2})}^{6}$

Step 4.48

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

$64 x^{6} - 192 x^{5} y^{2} + 240 x^{4} y^{4} - 160 x^{3} y^{6} + 60 x^{2} y^{8} - 12 x y^{10} + {(2 x)}^{0} {(- y^{2})}^{6}$

Step 4.49

Apply the product rule to $2 x$ .

$64 x^{6} - 192 x^{5} y^{2} + 240 x^{4} y^{4} - 160 x^{3} y^{6} + 60 x^{2} y^{8} - 12 x y^{10} + 2^{0} x^{0} {(- y^{2})}^{6}$

Step 4.50

Anything raised to $0$ is $1$ .

$64 x^{6} - 192 x^{5} y^{2} + 240 x^{4} y^{4} - 160 x^{3} y^{6} + 60 x^{2} y^{8} - 12 x y^{10} + 1 x^{0} {(- y^{2})}^{6}$

Step 4.51

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

$64 x^{6} - 192 x^{5} y^{2} + 240 x^{4} y^{4} - 160 x^{3} y^{6} + 60 x^{2} y^{8} - 12 x y^{10} + x^{0} {(- y^{2})}^{6}$

Step 4.52

Anything raised to $0$ is $1$ .

$64 x^{6} - 192 x^{5} y^{2} + 240 x^{4} y^{4} - 160 x^{3} y^{6} + 60 x^{2} y^{8} - 12 x y^{10} + 1 {(- y^{2})}^{6}$

Step 4.53

Multiply ${(- y^{2})}^{6}$ by $1$ .

$64 x^{6} - 192 x^{5} y^{2} + 240 x^{4} y^{4} - 160 x^{3} y^{6} + 60 x^{2} y^{8} - 12 x y^{10} + {(- y^{2})}^{6}$

Step 4.54

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

$64 x^{6} - 192 x^{5} y^{2} + 240 x^{4} y^{4} - 160 x^{3} y^{6} + 60 x^{2} y^{8} - 12 x y^{10} + {(- 1)}^{6} {(y^{2})}^{6}$

Step 4.55

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

$64 x^{6} - 192 x^{5} y^{2} + 240 x^{4} y^{4} - 160 x^{3} y^{6} + 60 x^{2} y^{8} - 12 x y^{10} + 1 {(y^{2})}^{6}$

Step 4.56

Multiply ${(y^{2})}^{6}$ by $1$ .

$64 x^{6} - 192 x^{5} y^{2} + 240 x^{4} y^{4} - 160 x^{3} y^{6} + 60 x^{2} y^{8} - 12 x y^{10} + {(y^{2})}^{6}$

Step 4.57

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

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Step 4.57.1

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

$64 x^{6} - 192 x^{5} y^{2} + 240 x^{4} y^{4} - 160 x^{3} y^{6} + 60 x^{2} y^{8} - 12 x y^{10} + y^{2 \cdot 6}$

Step 4.57.2

Multiply $2$ by $6$ .

$64 x^{6} - 192 x^{5} y^{2} + 240 x^{4} y^{4} - 160 x^{3} y^{6} + 60 x^{2} y^{8} - 12 x y^{10} + y^{12}$