Precalculus Examples

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

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 ${(2 x - 3 y)}^{3}$ , $n = 3$ 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 $a$ $2 x$ and $b$ $- 3 y$ into the expression.

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

Step 4

Simplify each term.

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Step 4.1

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

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

Step 4.2

Apply the product rule to $2 x$ .

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

Step 4.3

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

$8 x^{3} {(- 3 y)}^{0} + 3 {(2 x)}^{2} {(- 3 y)}^{1} + 3 {(2 x)}^{1} {(- 3 y)}^{2} + 1 {(2 x)}^{0} {(- 3 y)}^{3}$

Step 4.4

Apply the product rule to $- 3 y$ .

$8 x^{3} ({(- 3)}^{0} y^{0}) + 3 {(2 x)}^{2} {(- 3 y)}^{1} + 3 {(2 x)}^{1} {(- 3 y)}^{2} + 1 {(2 x)}^{0} {(- 3 y)}^{3}$

Step 4.5

Rewrite using the commutative property of multiplication.

$8 \cdot {(- 3)}^{0} x^{3} y^{0} + 3 {(2 x)}^{2} {(- 3 y)}^{1} + 3 {(2 x)}^{1} {(- 3 y)}^{2} + 1 {(2 x)}^{0} {(- 3 y)}^{3}$

Step 4.6

Anything raised to $0$ is $1$ .

$8 \cdot 1 x^{3} y^{0} + 3 {(2 x)}^{2} {(- 3 y)}^{1} + 3 {(2 x)}^{1} {(- 3 y)}^{2} + 1 {(2 x)}^{0} {(- 3 y)}^{3}$

Step 4.7

Multiply $8$ by $1$ .

$8 x^{3} y^{0} + 3 {(2 x)}^{2} {(- 3 y)}^{1} + 3 {(2 x)}^{1} {(- 3 y)}^{2} + 1 {(2 x)}^{0} {(- 3 y)}^{3}$

Step 4.8

Anything raised to $0$ is $1$ .

$8 x^{3} \cdot 1 + 3 {(2 x)}^{2} {(- 3 y)}^{1} + 3 {(2 x)}^{1} {(- 3 y)}^{2} + 1 {(2 x)}^{0} {(- 3 y)}^{3}$

Step 4.9

Multiply $8$ by $1$ .

$8 x^{3} + 3 {(2 x)}^{2} {(- 3 y)}^{1} + 3 {(2 x)}^{1} {(- 3 y)}^{2} + 1 {(2 x)}^{0} {(- 3 y)}^{3}$

Step 4.10

Apply the product rule to $2 x$ .

$8 x^{3} + 3 (2^{2} x^{2}) {(- 3 y)}^{1} + 3 {(2 x)}^{1} {(- 3 y)}^{2} + 1 {(2 x)}^{0} {(- 3 y)}^{3}$

Step 4.11

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

$8 x^{3} + 3 (4 x^{2}) {(- 3 y)}^{1} + 3 {(2 x)}^{1} {(- 3 y)}^{2} + 1 {(2 x)}^{0} {(- 3 y)}^{3}$

Step 4.12

Multiply $4$ by $3$ .

$8 x^{3} + 12 x^{2} {(- 3 y)}^{1} + 3 {(2 x)}^{1} {(- 3 y)}^{2} + 1 {(2 x)}^{0} {(- 3 y)}^{3}$

Step 4.13

Simplify.

$8 x^{3} + 12 x^{2} (- 3 y) + 3 {(2 x)}^{1} {(- 3 y)}^{2} + 1 {(2 x)}^{0} {(- 3 y)}^{3}$

Step 4.14

Rewrite using the commutative property of multiplication.

$8 x^{3} + 12 \cdot - 3 x^{2} y + 3 {(2 x)}^{1} {(- 3 y)}^{2} + 1 {(2 x)}^{0} {(- 3 y)}^{3}$

Step 4.15

Multiply $12$ by $- 3$ .

$8 x^{3} - 36 x^{2} y + 3 {(2 x)}^{1} {(- 3 y)}^{2} + 1 {(2 x)}^{0} {(- 3 y)}^{3}$

Step 4.16

Simplify.

$8 x^{3} - 36 x^{2} y + 3 (2 x) {(- 3 y)}^{2} + 1 {(2 x)}^{0} {(- 3 y)}^{3}$

Step 4.17

Multiply $2$ by $3$ .

$8 x^{3} - 36 x^{2} y + 6 x {(- 3 y)}^{2} + 1 {(2 x)}^{0} {(- 3 y)}^{3}$

Step 4.18

Apply the product rule to $- 3 y$ .

$8 x^{3} - 36 x^{2} y + 6 x ({(- 3)}^{2} y^{2}) + 1 {(2 x)}^{0} {(- 3 y)}^{3}$

Step 4.19

Rewrite using the commutative property of multiplication.

$8 x^{3} - 36 x^{2} y + 6 \cdot {(- 3)}^{2} x y^{2} + 1 {(2 x)}^{0} {(- 3 y)}^{3}$

Step 4.20

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

$8 x^{3} - 36 x^{2} y + 6 \cdot 9 x y^{2} + 1 {(2 x)}^{0} {(- 3 y)}^{3}$

Step 4.21

Multiply $6$ by $9$ .

$8 x^{3} - 36 x^{2} y + 54 x y^{2} + 1 {(2 x)}^{0} {(- 3 y)}^{3}$

Step 4.22

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

$8 x^{3} - 36 x^{2} y + 54 x y^{2} + {(2 x)}^{0} {(- 3 y)}^{3}$

Step 4.23

Apply the product rule to $2 x$ .

$8 x^{3} - 36 x^{2} y + 54 x y^{2} + 2^{0} x^{0} {(- 3 y)}^{3}$

Step 4.24

Anything raised to $0$ is $1$ .

$8 x^{3} - 36 x^{2} y + 54 x y^{2} + 1 x^{0} {(- 3 y)}^{3}$

Step 4.25

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

$8 x^{3} - 36 x^{2} y + 54 x y^{2} + x^{0} {(- 3 y)}^{3}$

Step 4.26

Anything raised to $0$ is $1$ .

$8 x^{3} - 36 x^{2} y + 54 x y^{2} + 1 {(- 3 y)}^{3}$

Step 4.27

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

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

Step 4.28

Apply the product rule to $- 3 y$ .

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

Step 4.29

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

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