Algebra Examples

{(3 x - y)}^{3}

${(3 x - y)}^{3}$

Step 1

Use the binomial expansion theorem to find each term. The binomial theorem states

{(a + b)}^{n} = n \sum k = 0 n C k \cdot (a^{n - k} b^{k})

${(a + b)}^{n} = \sum_{k = 0}^{n} n C k \cdot (a^{n - k} b^{k})$ .

3 \sum k = 0 \frac{3!}{(3 - k)! k!} \cdot {(3 x)}^{3 - k} \cdot {(- y)}^{k}

$\sum_{k = 0}^{3} \frac{3!}{(3 - k)! k!} \cdot {(3 x)}^{3 - k} \cdot {(- y)}^{k}$

Step 2

Expand the summation.

\frac{3!}{(3 - 0)! 0!} \cdot {(3 x)}^{3 - 0} \cdot {(- y)}^{0} + \frac{3!}{(3 - 1)! 1!} \cdot {(3 x)}^{3 - 1} \cdot {(- y)}^{1} + \frac{3!}{(3 - 2)! 2!} \cdot {(3 x)}^{3 - 2} \cdot {(- y)}^{2} + \frac{3!}{(3 - 3)! 3!} \cdot {(3 x)}^{3 - 3} \cdot {(- y)}^{3}

$\frac{3!}{(3 - 0)! 0!} \cdot {(3 x)}^{3 - 0} \cdot {(- y)}^{0} + \frac{3!}{(3 - 1)! 1!} \cdot {(3 x)}^{3 - 1} \cdot {(- y)}^{1} + \frac{3!}{(3 - 2)! 2!} \cdot {(3 x)}^{3 - 2} \cdot {(- y)}^{2} + \frac{3!}{(3 - 3)! 3!} \cdot {(3 x)}^{3 - 3} \cdot {(- y)}^{3}$

Step 3

Simplify the exponents for each term of the expansion.

1 \cdot {(3 x)}^{3} \cdot {(- y)}^{0} + 3 \cdot {(3 x)}^{2} \cdot {(- y)}^{1} + 3 \cdot {(3 x)}^{1} \cdot {(- y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- y)}^{3}

$1 \cdot {(3 x)}^{3} \cdot {(- y)}^{0} + 3 \cdot {(3 x)}^{2} \cdot {(- y)}^{1} + 3 \cdot {(3 x)}^{1} \cdot {(- y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- y)}^{3}$

Step 4

Simplify each term.

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Step 4.1

Multiply

{(3 x)}^{3}

${(3 x)}^{3}$ by

1

$1$ .

{(3 x)}^{3} \cdot {(- y)}^{0} + 3 \cdot {(3 x)}^{2} \cdot {(- y)}^{1} + 3 \cdot {(3 x)}^{1} \cdot {(- y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- y)}^{3}

${(3 x)}^{3} \cdot {(- y)}^{0} + 3 \cdot {(3 x)}^{2} \cdot {(- y)}^{1} + 3 \cdot {(3 x)}^{1} \cdot {(- y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- y)}^{3}$

Step 4.2

Apply the product rule to

3 x

$3 x$ .

3^{3} x^{3} \cdot {(- y)}^{0} + 3 \cdot {(3 x)}^{2} \cdot {(- y)}^{1} + 3 \cdot {(3 x)}^{1} \cdot {(- y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- y)}^{3}

$3^{3} x^{3} \cdot {(- y)}^{0} + 3 \cdot {(3 x)}^{2} \cdot {(- y)}^{1} + 3 \cdot {(3 x)}^{1} \cdot {(- y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- y)}^{3}$

Step 4.3

Raise

3

$3$ to the power of

3

$3$ .

27 x^{3} \cdot {(- y)}^{0} + 3 \cdot {(3 x)}^{2} \cdot {(- y)}^{1} + 3 \cdot {(3 x)}^{1} \cdot {(- y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- y)}^{3}

$27 x^{3} \cdot {(- y)}^{0} + 3 \cdot {(3 x)}^{2} \cdot {(- y)}^{1} + 3 \cdot {(3 x)}^{1} \cdot {(- y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- y)}^{3}$

Step 4.4

Apply the product rule to

- y

$- y$ .

27 x^{3} \cdot ({(- 1)}^{0} y^{0}) + 3 \cdot {(3 x)}^{2} \cdot {(- y)}^{1} + 3 \cdot {(3 x)}^{1} \cdot {(- y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- y)}^{3}

$27 x^{3} \cdot ({(- 1)}^{0} y^{0}) + 3 \cdot {(3 x)}^{2} \cdot {(- y)}^{1} + 3 \cdot {(3 x)}^{1} \cdot {(- y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- y)}^{3}$

Step 4.5

Rewrite using the commutative property of multiplication.

27 \cdot {(- 1)}^{0} x^{3} y^{0} + 3 \cdot {(3 x)}^{2} \cdot {(- y)}^{1} + 3 \cdot {(3 x)}^{1} \cdot {(- y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- y)}^{3}

$27 \cdot {(- 1)}^{0} x^{3} y^{0} + 3 \cdot {(3 x)}^{2} \cdot {(- y)}^{1} + 3 \cdot {(3 x)}^{1} \cdot {(- y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- y)}^{3}$

Step 4.6

Anything raised to

0

$0$ is

1

$1$ .

27 \cdot 1 x^{3} y^{0} + 3 \cdot {(3 x)}^{2} \cdot {(- y)}^{1} + 3 \cdot {(3 x)}^{1} \cdot {(- y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- y)}^{3}

$27 \cdot 1 x^{3} y^{0} + 3 \cdot {(3 x)}^{2} \cdot {(- y)}^{1} + 3 \cdot {(3 x)}^{1} \cdot {(- y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- y)}^{3}$

Step 4.7

Multiply

27

$27$ by

1

$1$ .

27 x^{3} y^{0} + 3 \cdot {(3 x)}^{2} \cdot {(- y)}^{1} + 3 \cdot {(3 x)}^{1} \cdot {(- y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- y)}^{3}

$27 x^{3} y^{0} + 3 \cdot {(3 x)}^{2} \cdot {(- y)}^{1} + 3 \cdot {(3 x)}^{1} \cdot {(- y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- y)}^{3}$

Step 4.8

Anything raised to

0

$0$ is

1

$1$ .

27 x^{3} \cdot 1 + 3 \cdot {(3 x)}^{2} \cdot {(- y)}^{1} + 3 \cdot {(3 x)}^{1} \cdot {(- y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- y)}^{3}

$27 x^{3} \cdot 1 + 3 \cdot {(3 x)}^{2} \cdot {(- y)}^{1} + 3 \cdot {(3 x)}^{1} \cdot {(- y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- y)}^{3}$

Step 4.9

Multiply

27

$27$ by

1

$1$ .

27 x^{3} + 3 \cdot {(3 x)}^{2} \cdot {(- y)}^{1} + 3 \cdot {(3 x)}^{1} \cdot {(- y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- y)}^{3}

$27 x^{3} + 3 \cdot {(3 x)}^{2} \cdot {(- y)}^{1} + 3 \cdot {(3 x)}^{1} \cdot {(- y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- y)}^{3}$

Step 4.10

Apply the product rule to

3 x

$3 x$ .

27 x^{3} + 3 \cdot (3^{2} x^{2}) \cdot {(- y)}^{1} + 3 \cdot {(3 x)}^{1} \cdot {(- y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- y)}^{3}

$27 x^{3} + 3 \cdot (3^{2} x^{2}) \cdot {(- y)}^{1} + 3 \cdot {(3 x)}^{1} \cdot {(- y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- y)}^{3}$

Step 4.11

Multiply

3

$3$ by

3^{2}

$3^{2}$ by adding the exponents.

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Step 4.11.1

Move

3^{2}

$3^{2}$ .

27 x^{3} + 3^{2} \cdot 3 \cdot x^{2} \cdot {(- y)}^{1} + 3 \cdot {(3 x)}^{1} \cdot {(- y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- y)}^{3}

$27 x^{3} + 3^{2} \cdot 3 \cdot x^{2} \cdot {(- y)}^{1} + 3 \cdot {(3 x)}^{1} \cdot {(- y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- y)}^{3}$

Step 4.11.2

Multiply

3^{2}

$3^{2}$ by

3

$3$ .

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Step 4.11.2.1

Raise

3

$3$ to the power of

1

$1$ .

27 x^{3} + 3^{2} \cdot 3^{1} \cdot x^{2} \cdot {(- y)}^{1} + 3 \cdot {(3 x)}^{1} \cdot {(- y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- y)}^{3}

$27 x^{3} + 3^{2} \cdot 3^{1} \cdot x^{2} \cdot {(- y)}^{1} + 3 \cdot {(3 x)}^{1} \cdot {(- y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- y)}^{3}$

Step 4.11.2.2

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

27 x^{3} + 3^{2 + 1} \cdot x^{2} \cdot {(- y)}^{1} + 3 \cdot {(3 x)}^{1} \cdot {(- y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- y)}^{3}

$27 x^{3} + 3^{2 + 1} \cdot x^{2} \cdot {(- y)}^{1} + 3 \cdot {(3 x)}^{1} \cdot {(- y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- y)}^{3}$

27 x^{3} + 3^{2 + 1} \cdot x^{2} \cdot {(- y)}^{1} + 3 \cdot {(3 x)}^{1} \cdot {(- y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- y)}^{3}

$27 x^{3} + 3^{2 + 1} \cdot x^{2} \cdot {(- y)}^{1} + 3 \cdot {(3 x)}^{1} \cdot {(- y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- y)}^{3}$

Step 4.11.3

Add

2

$2$ and

1

$1$ .

27 x^{3} + 3^{3} \cdot x^{2} \cdot {(- y)}^{1} + 3 \cdot {(3 x)}^{1} \cdot {(- y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- y)}^{3}

$27 x^{3} + 3^{3} \cdot x^{2} \cdot {(- y)}^{1} + 3 \cdot {(3 x)}^{1} \cdot {(- y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- y)}^{3}$

27 x^{3} + 3^{3} \cdot x^{2} \cdot {(- y)}^{1} + 3 \cdot {(3 x)}^{1} \cdot {(- y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- y)}^{3}

$27 x^{3} + 3^{3} \cdot x^{2} \cdot {(- y)}^{1} + 3 \cdot {(3 x)}^{1} \cdot {(- y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- y)}^{3}$

Step 4.12

Simplify

3^{3} \cdot x^{2} \cdot {(- y)}^{1}

$3^{3} \cdot x^{2} \cdot {(- y)}^{1}$ .

27 x^{3} + 3^{3} \cdot x^{2} \cdot (- y) + 3 \cdot {(3 x)}^{1} \cdot {(- y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- y)}^{3}

$27 x^{3} + 3^{3} \cdot x^{2} \cdot (- y) + 3 \cdot {(3 x)}^{1} \cdot {(- y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- y)}^{3}$

Step 4.13

Rewrite using the commutative property of multiplication.

27 x^{3} + 3^{3} \cdot - 1 x^{2} y + 3 \cdot {(3 x)}^{1} \cdot {(- y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- y)}^{3}

$27 x^{3} + 3^{3} \cdot - 1 x^{2} y + 3 \cdot {(3 x)}^{1} \cdot {(- y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- y)}^{3}$

Step 4.14

Raise

3

$3$ to the power of

3

$3$ .

27 x^{3} + 27 \cdot - 1 x^{2} y + 3 \cdot {(3 x)}^{1} \cdot {(- y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- y)}^{3}

$27 x^{3} + 27 \cdot - 1 x^{2} y + 3 \cdot {(3 x)}^{1} \cdot {(- y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- y)}^{3}$

Step 4.15

Multiply

27

$27$ by

- 1

$- 1$ .

27 x^{3} - 27 x^{2} y + 3 \cdot {(3 x)}^{1} \cdot {(- y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- y)}^{3}

$27 x^{3} - 27 x^{2} y + 3 \cdot {(3 x)}^{1} \cdot {(- y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- y)}^{3}$

Step 4.16

Simplify.

27 x^{3} - 27 x^{2} y + 3 \cdot (3 x) \cdot {(- y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- y)}^{3}

$27 x^{3} - 27 x^{2} y + 3 \cdot (3 x) \cdot {(- y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- y)}^{3}$

Step 4.17

Multiply

3

$3$ by

3

$3$ .

27 x^{3} - 27 x^{2} y + 9 x \cdot {(- y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- y)}^{3}

$27 x^{3} - 27 x^{2} y + 9 x \cdot {(- y)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- y)}^{3}$

Step 4.18

Apply the product rule to

- y

$- y$ .

27 x^{3} - 27 x^{2} y + 9 x \cdot ({(- 1)}^{2} y^{2}) + 1 \cdot {(3 x)}^{0} \cdot {(- y)}^{3}

$27 x^{3} - 27 x^{2} y + 9 x \cdot ({(- 1)}^{2} y^{2}) + 1 \cdot {(3 x)}^{0} \cdot {(- y)}^{3}$

Step 4.19

Rewrite using the commutative property of multiplication.

27 x^{3} - 27 x^{2} y + 9 \cdot {(- 1)}^{2} x y^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- y)}^{3}

$27 x^{3} - 27 x^{2} y + 9 \cdot {(- 1)}^{2} x y^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- y)}^{3}$

Step 4.20

Raise

- 1

$- 1$ to the power of

2

$2$ .

27 x^{3} - 27 x^{2} y + 9 \cdot 1 x y^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- y)}^{3}

$27 x^{3} - 27 x^{2} y + 9 \cdot 1 x y^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- y)}^{3}$

Step 4.21

Multiply

9

$9$ by

1

$1$ .

27 x^{3} - 27 x^{2} y + 9 x y^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- y)}^{3}

$27 x^{3} - 27 x^{2} y + 9 x y^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- y)}^{3}$

Step 4.22

Multiply

{(3 x)}^{0}

${(3 x)}^{0}$ by

1

$1$ .

27 x^{3} - 27 x^{2} y + 9 x y^{2} + {(3 x)}^{0} \cdot {(- y)}^{3}

$27 x^{3} - 27 x^{2} y + 9 x y^{2} + {(3 x)}^{0} \cdot {(- y)}^{3}$

Step 4.23

Apply the product rule to

3 x

$3 x$ .

27 x^{3} - 27 x^{2} y + 9 x y^{2} + 3^{0} x^{0} \cdot {(- y)}^{3}

$27 x^{3} - 27 x^{2} y + 9 x y^{2} + 3^{0} x^{0} \cdot {(- y)}^{3}$

Step 4.24

Anything raised to

$0$ is

$1$ .

$27 x^{3} - 27 x^{2} y + 9 x y^{2} + 1 x^{0} \cdot {(- y)}^{3}$

Step 4.25

Multiply

$x^{0}$ by

$1$ .

$27 x^{3} - 27 x^{2} y + 9 x y^{2} + x^{0} \cdot {(- y)}^{3}$

Step 4.26

Anything raised to

$0$ is

$1$ .

$27 x^{3} - 27 x^{2} y + 9 x y^{2} + 1 \cdot {(- y)}^{3}$

Step 4.27

Multiply

${(- y)}^{3}$ by

$1$ .

$27 x^{3} - 27 x^{2} y + 9 x y^{2} + {(- y)}^{3}$

Step 4.28

Apply the product rule to

$- y$ .

$27 x^{3} - 27 x^{2} y + 9 x y^{2} + {(- 1)}^{3} y^{3}$

Step 4.29

Raise

$- 1$ to the power of

$3$ .

$27 x^{3} - 27 x^{2} y + 9 x y^{2} - y^{3}$