Algebra Examples

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

Step 1

Use the binomial expansion theorem to find each term. The binomial theorem states ${(a + b)}^{n} = \sum_{k = 0}^{n} n C k \cdot (a^{n - k} b^{k})$ .

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

Step 2

Expand the summation.

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

Step 3

Simplify the exponents for each term of the expansion.

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

Step 4

Simplify each term.

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Step 4.1

Multiply ${(a x)}^{3}$ by $1$ .

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

Step 4.2

Apply the product rule to $a x$ .

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

Step 4.3

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.3.1

Apply the product rule to $- b y$ .

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

Step 4.3.2

Apply the product rule to $- b$ .

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

Step 4.4

Rewrite using the commutative property of multiplication.

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

Step 4.5

Anything raised to $0$ is $1$ .

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

Step 4.6

Multiply $a^{3}$ by $1$ .

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

Step 4.7

Anything raised to $0$ is $1$ .

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

Step 4.8

Multiply $a^{3}$ by $1$ .

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

Step 4.9

Anything raised to $0$ is $1$ .

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

Step 4.10

Multiply $a^{3}$ by $1$ .

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

Step 4.11

Apply the product rule to $a x$ .

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

Step 4.12

Simplify.

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

Step 4.13

Multiply $- 1$ by $3$ .

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

Step 4.14

Simplify.

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

Step 4.15

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.15.1

Apply the product rule to $- b y$ .

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

Step 4.15.2

Apply the product rule to $- b$ .

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

Step 4.16

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

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

Step 4.17

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

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

Step 4.18

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

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

Step 4.19

Apply the product rule to $a x$ .

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

Step 4.20

Anything raised to $0$ is $1$ .

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

Step 4.21

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

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

Step 4.22

Anything raised to $0$ is $1$ .

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

Step 4.23

Multiply ${(- b y)}^{3}$ by $1$ .

$a^{3} x^{3} - 3 a^{2} x^{2} b y + 3 a x b^{2} y^{2} + {(- b y)}^{3}$

Step 4.24

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.24.1

Apply the product rule to $- b y$ .

$a^{3} x^{3} - 3 a^{2} x^{2} b y + 3 a x b^{2} y^{2} + {(- b)}^{3} y^{3}$

Step 4.24.2

Apply the product rule to $- b$ .

$a^{3} x^{3} - 3 a^{2} x^{2} b y + 3 a x b^{2} y^{2} + {(- 1)}^{3} b^{3} y^{3}$

Step 4.25

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

$a^{3} x^{3} - 3 a^{2} x^{2} b y + 3 a x b^{2} y^{2} - b^{3} y^{3}$