Finite Math Examples

${(1 + 2 i)}^{3}$

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

${(1 + 2 i)}^{3}$ ,

$n = 3$ 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

$a$

$1$ and

$b$

$2 i$ into the expression.

$1 {(1)}^{3} {(2 i)}^{0} + 3 {(1)}^{2} {(2 i)}^{1} + 3 {(1)}^{1} {(2 i)}^{2} + 1 {(1)}^{0} {(2 i)}^{3}$

Step 4

Simplify the expression.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply

$1$ by

${(1)}^{3}$ by adding the exponents.

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Step 4.1.1.1

Multiply

$1$ by

${(1)}^{3}$ .

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Step 4.1.1.1.1

Raise

$1$ to the power of

$1$ .

$1^{1} {(1)}^{3} {(2 i)}^{0} + 3 {(1)}^{2} {(2 i)}^{1} + 3 {(1)}^{1} {(2 i)}^{2} + 1 {(1)}^{0} {(2 i)}^{3}$

Step 4.1.1.1.2

Use the power rule

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

$1^{1 + 3} {(2 i)}^{0} + 3 {(1)}^{2} {(2 i)}^{1} + 3 {(1)}^{1} {(2 i)}^{2} + 1 {(1)}^{0} {(2 i)}^{3}$

Step 4.1.1.2

Add

$1$ and

$3$ .

$1^{4} {(2 i)}^{0} + 3 {(1)}^{2} {(2 i)}^{1} + 3 {(1)}^{1} {(2 i)}^{2} + 1 {(1)}^{0} {(2 i)}^{3}$

Step 4.1.2

Simplify

$1^{4} {(2 i)}^{0}$ .

$1^{4} + 3 {(1)}^{2} {(2 i)}^{1} + 3 {(1)}^{1} {(2 i)}^{2} + 1 {(1)}^{0} {(2 i)}^{3}$

Step 4.1.3

One to any power is one.

$1 + 3 {(1)}^{2} {(2 i)}^{1} + 3 {(1)}^{1} {(2 i)}^{2} + 1 {(1)}^{0} {(2 i)}^{3}$

Step 4.1.4

One to any power is one.

$1 + 3 \cdot 1 {(2 i)}^{1} + 3 {(1)}^{1} {(2 i)}^{2} + 1 {(1)}^{0} {(2 i)}^{3}$

Step 4.1.5

Multiply

$3$ by

$1$ .

$1 + 3 {(2 i)}^{1} + 3 {(1)}^{1} {(2 i)}^{2} + 1 {(1)}^{0} {(2 i)}^{3}$

Step 4.1.6

Simplify.

$1 + 3 (2 i) + 3 {(1)}^{1} {(2 i)}^{2} + 1 {(1)}^{0} {(2 i)}^{3}$

Step 4.1.7

Multiply

$2$ by

$3$ .

$1 + 6 i + 3 {(1)}^{1} {(2 i)}^{2} + 1 {(1)}^{0} {(2 i)}^{3}$

Step 4.1.8

Evaluate the exponent.

$1 + 6 i + 3 \cdot 1 {(2 i)}^{2} + 1 {(1)}^{0} {(2 i)}^{3}$

Step 4.1.9

Multiply

$3$ by

$1$ .

$1 + 6 i + 3 {(2 i)}^{2} + 1 {(1)}^{0} {(2 i)}^{3}$

Step 4.1.10

Apply the product rule to

$2 i$ .

$1 + 6 i + 3 (2^{2} i^{2}) + 1 {(1)}^{0} {(2 i)}^{3}$

Step 4.1.11

Raise

$2$ to the power of

$2$ .

$1 + 6 i + 3 (4 i^{2}) + 1 {(1)}^{0} {(2 i)}^{3}$

Step 4.1.12

Rewrite

$i^{2}$ as

$- 1$ .

$1 + 6 i + 3 (4 \cdot - 1) + 1 {(1)}^{0} {(2 i)}^{3}$

Step 4.1.13

Multiply

$3 (4 \cdot - 1)$ .

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Step 4.1.13.1

Multiply

$4$ by

$- 1$ .

$1 + 6 i + 3 \cdot - 4 + 1 {(1)}^{0} {(2 i)}^{3}$

Step 4.1.13.2

Multiply

$3$ by

$- 4$ .

$1 + 6 i - 12 + 1 {(1)}^{0} {(2 i)}^{3}$

Step 4.1.14

Multiply

$1$ by

${(1)}^{0}$ by adding the exponents.

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Step 4.1.14.1

Multiply

$1$ by

${(1)}^{0}$ .

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Step 4.1.14.1.1

Raise

$1$ to the power of

$1$ .

$1 + 6 i - 12 + 1^{1} {(1)}^{0} {(2 i)}^{3}$

Step 4.1.14.1.2

Use the power rule

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

$1 + 6 i - 12 + 1^{1 + 0} {(2 i)}^{3}$

Step 4.1.14.2

Add

$1$ and

$0$ .

$1 + 6 i - 12 + 1^{1} {(2 i)}^{3}$

Step 4.1.15

Simplify

$1^{1} {(2 i)}^{3}$ .

$1 + 6 i - 12 + {(2 i)}^{3}$

Step 4.1.16

Apply the product rule to

$2 i$ .

$1 + 6 i - 12 + 2^{3} i^{3}$

Step 4.1.17

Raise

$2$ to the power of

$3$ .

$1 + 6 i - 12 + 8 i^{3}$

Step 4.1.18

Factor out

$i^{2}$ .

$1 + 6 i - 12 + 8 (i^{2} \cdot i)$

Step 4.1.19

Rewrite

$i^{2}$ as

$- 1$ .

$1 + 6 i - 12 + 8 (- 1 \cdot i)$

Step 4.1.20

Rewrite

$- 1 i$ as

$- i$ .

$1 + 6 i - 12 + 8 (- i)$

Step 4.1.21

Multiply

$- 1$ by

$8$ .

$1 + 6 i - 12 - 8 i$

Step 4.2

Simplify by adding terms.

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Step 4.2.1

Subtract

$12$ from

$1$ .

$- 11 + 6 i - 8 i$

Step 4.2.2

Subtract

$8 i$ from

$6 i$ .

$- 11 - 2 i$