Precalculus Examples

${(4 a - b)}^{3}$

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 ${(4 a - b)}^{3}$ , $n = 3$ 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 $a$ $4 a$ and $b$ $- b$ into the expression.

$1 {(4 a)}^{3} {(- b)}^{0} + 3 {(4 a)}^{2} {(- b)}^{1} + 3 {(4 a)}^{1} {(- b)}^{2} + 1 {(4 a)}^{0} {(- b)}^{3}$

Step 4

Simplify each term.

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Step 4.1

Multiply ${(4 a)}^{3}$ by $1$ .

${(4 a)}^{3} {(- b)}^{0} + 3 {(4 a)}^{2} {(- b)}^{1} + 3 {(4 a)}^{1} {(- b)}^{2} + 1 {(4 a)}^{0} {(- b)}^{3}$

Step 4.2

Apply the product rule to $4 a$ .

$4^{3} a^{3} {(- b)}^{0} + 3 {(4 a)}^{2} {(- b)}^{1} + 3 {(4 a)}^{1} {(- b)}^{2} + 1 {(4 a)}^{0} {(- b)}^{3}$

Step 4.3

Raise $4$ to the power of $3$ .

$64 a^{3} {(- b)}^{0} + 3 {(4 a)}^{2} {(- b)}^{1} + 3 {(4 a)}^{1} {(- b)}^{2} + 1 {(4 a)}^{0} {(- b)}^{3}$

Step 4.4

Apply the product rule to $- b$ .

$64 a^{3} ({(- 1)}^{0} b^{0}) + 3 {(4 a)}^{2} {(- b)}^{1} + 3 {(4 a)}^{1} {(- b)}^{2} + 1 {(4 a)}^{0} {(- b)}^{3}$

Step 4.5

Rewrite using the commutative property of multiplication.

$64 \cdot {(- 1)}^{0} a^{3} b^{0} + 3 {(4 a)}^{2} {(- b)}^{1} + 3 {(4 a)}^{1} {(- b)}^{2} + 1 {(4 a)}^{0} {(- b)}^{3}$

Step 4.6

Anything raised to $0$ is $1$ .

$64 \cdot 1 a^{3} b^{0} + 3 {(4 a)}^{2} {(- b)}^{1} + 3 {(4 a)}^{1} {(- b)}^{2} + 1 {(4 a)}^{0} {(- b)}^{3}$

Step 4.7

Multiply $64$ by $1$ .

$64 a^{3} b^{0} + 3 {(4 a)}^{2} {(- b)}^{1} + 3 {(4 a)}^{1} {(- b)}^{2} + 1 {(4 a)}^{0} {(- b)}^{3}$

Step 4.8

Anything raised to $0$ is $1$ .

$64 a^{3} \cdot 1 + 3 {(4 a)}^{2} {(- b)}^{1} + 3 {(4 a)}^{1} {(- b)}^{2} + 1 {(4 a)}^{0} {(- b)}^{3}$

Step 4.9

Multiply $64$ by $1$ .

$64 a^{3} + 3 {(4 a)}^{2} {(- b)}^{1} + 3 {(4 a)}^{1} {(- b)}^{2} + 1 {(4 a)}^{0} {(- b)}^{3}$

Step 4.10

Apply the product rule to $4 a$ .

$64 a^{3} + 3 (4^{2} a^{2}) {(- b)}^{1} + 3 {(4 a)}^{1} {(- b)}^{2} + 1 {(4 a)}^{0} {(- b)}^{3}$

Step 4.11

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

$64 a^{3} + 3 (16 a^{2}) {(- b)}^{1} + 3 {(4 a)}^{1} {(- b)}^{2} + 1 {(4 a)}^{0} {(- b)}^{3}$

Step 4.12

Multiply $16$ by $3$ .

$64 a^{3} + 48 a^{2} {(- b)}^{1} + 3 {(4 a)}^{1} {(- b)}^{2} + 1 {(4 a)}^{0} {(- b)}^{3}$

Step 4.13

Simplify.

$64 a^{3} + 48 a^{2} (- b) + 3 {(4 a)}^{1} {(- b)}^{2} + 1 {(4 a)}^{0} {(- b)}^{3}$

Step 4.14

Rewrite using the commutative property of multiplication.

$64 a^{3} + 48 \cdot - 1 a^{2} b + 3 {(4 a)}^{1} {(- b)}^{2} + 1 {(4 a)}^{0} {(- b)}^{3}$

Step 4.15

Multiply $48$ by $- 1$ .

$64 a^{3} - 48 a^{2} b + 3 {(4 a)}^{1} {(- b)}^{2} + 1 {(4 a)}^{0} {(- b)}^{3}$

Step 4.16

Simplify.

$64 a^{3} - 48 a^{2} b + 3 (4 a) {(- b)}^{2} + 1 {(4 a)}^{0} {(- b)}^{3}$

Step 4.17

Multiply $4$ by $3$ .

$64 a^{3} - 48 a^{2} b + 12 a {(- b)}^{2} + 1 {(4 a)}^{0} {(- b)}^{3}$

Step 4.18

Apply the product rule to $- b$ .

$64 a^{3} - 48 a^{2} b + 12 a ({(- 1)}^{2} b^{2}) + 1 {(4 a)}^{0} {(- b)}^{3}$

Step 4.19

Rewrite using the commutative property of multiplication.

$64 a^{3} - 48 a^{2} b + 12 \cdot {(- 1)}^{2} a b^{2} + 1 {(4 a)}^{0} {(- b)}^{3}$

Step 4.20

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

$64 a^{3} - 48 a^{2} b + 12 \cdot 1 a b^{2} + 1 {(4 a)}^{0} {(- b)}^{3}$

Step 4.21

Multiply $12$ by $1$ .

$64 a^{3} - 48 a^{2} b + 12 a b^{2} + 1 {(4 a)}^{0} {(- b)}^{3}$

Step 4.22

Multiply ${(4 a)}^{0}$ by $1$ .

$64 a^{3} - 48 a^{2} b + 12 a b^{2} + {(4 a)}^{0} {(- b)}^{3}$

Step 4.23

Apply the product rule to $4 a$ .

$64 a^{3} - 48 a^{2} b + 12 a b^{2} + 4^{0} a^{0} {(- b)}^{3}$

Step 4.24

Anything raised to $0$ is $1$ .

$64 a^{3} - 48 a^{2} b + 12 a b^{2} + 1 a^{0} {(- b)}^{3}$

Step 4.25

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

$64 a^{3} - 48 a^{2} b + 12 a b^{2} + a^{0} {(- b)}^{3}$

Step 4.26

Anything raised to $0$ is $1$ .

$64 a^{3} - 48 a^{2} b + 12 a b^{2} + 1 {(- b)}^{3}$

Step 4.27

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

$64 a^{3} - 48 a^{2} b + 12 a b^{2} + {(- b)}^{3}$

Step 4.28

Apply the product rule to $- b$ .

$64 a^{3} - 48 a^{2} b + 12 a b^{2} + {(- 1)}^{3} b^{3}$

Step 4.29

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

$64 a^{3} - 48 a^{2} b + 12 a b^{2} - b^{3}$