Precalculus Examples

${(3 x - 2 y)}^{3}$

Step 1

Pascal's Triangle can be displayed as such:

$1$

$1 - 1$

$1 - 2 - 1$

$1 - 3 - 3 - 1$

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 ${(3 x - 2 y)}^{3}$ , $n = 3$ so the coefficients of the expansion will correspond with line $4$ .

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 $1 - 3 - 3 - 1$ .

$1 a^{3} b^{0} + 3 a^{2} b + 3 a b^{2} + 1 a^{0} b^{3}$

Step 3

Substitute the actual values of $a$ $3 x$ and $b$ $- 2 y$ into the expression.

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

Step 4

Simplify each term.

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Step 4.1

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

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

Step 4.2

Apply the product rule to $3 x$ .

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

Step 4.3

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

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

Step 4.4

Apply the product rule to $- 2 y$ .

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

Step 4.5

Rewrite using the commutative property of multiplication.

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

Step 4.6

Anything raised to $0$ is $1$ .

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

Step 4.7

Multiply $27$ by $1$ .

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

Step 4.8

Anything raised to $0$ is $1$ .

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

Step 4.9

Multiply $27$ by $1$ .

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

Step 4.10

Apply the product rule to $3 x$ .

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

Step 4.11

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

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Step 4.11.1

Move $3^{2}$ .

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

Step 4.11.2

Multiply $3^{2}$ by $3$ .

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Step 4.11.2.1

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

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

Step 4.11.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 4.11.3

Add $2$ and $1$ .

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

Step 4.12

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

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

Step 4.13

Rewrite using the commutative property of multiplication.

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

Step 4.14

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

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

Step 4.15

Multiply $27$ by $- 2$ .

$27 x^{3} - 54 x^{2} y + 3 {(3 x)}^{1} {(- 2 y)}^{2} + 1 {(3 x)}^{0} {(- 2 y)}^{3}$

Step 4.16

Simplify.

$27 x^{3} - 54 x^{2} y + 3 (3 x) {(- 2 y)}^{2} + 1 {(3 x)}^{0} {(- 2 y)}^{3}$

Step 4.17

Multiply $3$ by $3$ .

$27 x^{3} - 54 x^{2} y + 9 x {(- 2 y)}^{2} + 1 {(3 x)}^{0} {(- 2 y)}^{3}$

Step 4.18

Apply the product rule to $- 2 y$ .

$27 x^{3} - 54 x^{2} y + 9 x ({(- 2)}^{2} y^{2}) + 1 {(3 x)}^{0} {(- 2 y)}^{3}$

Step 4.19

Rewrite using the commutative property of multiplication.

$27 x^{3} - 54 x^{2} y + 9 \cdot {(- 2)}^{2} x y^{2} + 1 {(3 x)}^{0} {(- 2 y)}^{3}$

Step 4.20

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

$27 x^{3} - 54 x^{2} y + 9 \cdot 4 x y^{2} + 1 {(3 x)}^{0} {(- 2 y)}^{3}$

Step 4.21

Multiply $9$ by $4$ .

$27 x^{3} - 54 x^{2} y + 36 x y^{2} + 1 {(3 x)}^{0} {(- 2 y)}^{3}$

Step 4.22

Multiply ${(3 x)}^{0}$ by $1$ .

$27 x^{3} - 54 x^{2} y + 36 x y^{2} + {(3 x)}^{0} {(- 2 y)}^{3}$

Step 4.23

Apply the product rule to $3 x$ .

$27 x^{3} - 54 x^{2} y + 36 x y^{2} + 3^{0} x^{0} {(- 2 y)}^{3}$

Step 4.24

Anything raised to $0$ is $1$ .

$27 x^{3} - 54 x^{2} y + 36 x y^{2} + 1 x^{0} {(- 2 y)}^{3}$

Step 4.25

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

$27 x^{3} - 54 x^{2} y + 36 x y^{2} + x^{0} {(- 2 y)}^{3}$

Step 4.26

Anything raised to $0$ is $1$ .

$27 x^{3} - 54 x^{2} y + 36 x y^{2} + 1 {(- 2 y)}^{3}$

Step 4.27

Multiply ${(- 2 y)}^{3}$ by $1$ .

$27 x^{3} - 54 x^{2} y + 36 x y^{2} + {(- 2 y)}^{3}$

Step 4.28

Apply the product rule to $- 2 y$ .

$27 x^{3} - 54 x^{2} y + 36 x y^{2} + {(- 2)}^{3} y^{3}$

Step 4.29

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

$27 x^{3} - 54 x^{2} y + 36 x y^{2} - 8 y^{3}$