Basic Math Examples

${(1 + \frac{2 i}{n})}^{3}$

Step 1

Use the Binomial Theorem.

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

Step 2

Simplify each term.

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Step 2.1

One to any power is one.

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

Step 2.2

One to any power is one.

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

Step 2.3

Multiply $3$ by $1$ .

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

Step 2.4

Multiply $3 \frac{2 i}{n}$ .

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Step 2.4.1

Combine $3$ and $\frac{2 i}{n}$ .

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

Step 2.4.2

Multiply $2$ by $3$ .

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

Step 2.5

Multiply $3$ by $1$ .

$1 + \frac{6 i}{n} + 3 {(\frac{2 i}{n})}^{2} + {(\frac{2 i}{n})}^{3}$

Step 2.6

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

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Step 2.6.1

Apply the product rule to $\frac{2 i}{n}$ .

$1 + \frac{6 i}{n} + 3 \frac{{(2 i)}^{2}}{n^{2}} + {(\frac{2 i}{n})}^{3}$

Step 2.6.2

Apply the product rule to $2 i$ .

$1 + \frac{6 i}{n} + 3 \frac{2^{2} i^{2}}{n^{2}} + {(\frac{2 i}{n})}^{3}$

Step 2.7

Simplify the numerator.

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Step 2.7.1

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

$1 + \frac{6 i}{n} + 3 \frac{4 i^{2}}{n^{2}} + {(\frac{2 i}{n})}^{3}$

Step 2.7.2

Rewrite $i^{2}$ as $- 1$ .

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

Step 2.8

Multiply $4$ by $- 1$ .

$1 + \frac{6 i}{n} + 3 \frac{- 4}{n^{2}} + {(\frac{2 i}{n})}^{3}$

Step 2.9

Move the negative in front of the fraction.

$1 + \frac{6 i}{n} + 3 (- \frac{4}{n^{2}}) + {(\frac{2 i}{n})}^{3}$

Step 2.10

Multiply $3 (- \frac{4}{n^{2}})$ .

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Step 2.10.1

Multiply $- 1$ by $3$ .

$1 + \frac{6 i}{n} - 3 \frac{4}{n^{2}} + {(\frac{2 i}{n})}^{3}$

Step 2.10.2

Combine $- 3$ and $\frac{4}{n^{2}}$ .

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

Step 2.10.3

Multiply $- 3$ by $4$ .

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

Step 2.11

Move the negative in front of the fraction.

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

Step 2.12

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

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Step 2.12.1

Apply the product rule to $\frac{2 i}{n}$ .

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

Step 2.12.2

Apply the product rule to $2 i$ .

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

Step 2.13

Simplify the numerator.

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Step 2.13.1

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

$1 + \frac{6 i}{n} - \frac{12}{n^{2}} + \frac{8 i^{3}}{n^{3}}$

Step 2.13.2

Factor out $i^{2}$ .

$1 + \frac{6 i}{n} - \frac{12}{n^{2}} + \frac{8 (i^{2} \cdot i)}{n^{3}}$

Step 2.13.3

Rewrite $i^{2}$ as $- 1$ .

$1 + \frac{6 i}{n} - \frac{12}{n^{2}} + \frac{8 (- 1 \cdot i)}{n^{3}}$

Step 2.13.4

Rewrite $- 1 i$ as $- i$ .

$1 + \frac{6 i}{n} - \frac{12}{n^{2}} + \frac{8 \cdot - 1 i}{n^{3}}$

Step 2.13.5

Combine exponents.

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Step 2.13.5.1

Factor out negative.

$1 + \frac{6 i}{n} - \frac{12}{n^{2}} + \frac{- (8 i)}{n^{3}}$

Step 2.13.5.2

Multiply $8$ by $- 1$ .

$1 + \frac{6 i}{n} - \frac{12}{n^{2}} + \frac{- 8 i}{n^{3}}$

Step 2.14

Move the negative in front of the fraction.

$1 + \frac{6 i}{n} - \frac{12}{n^{2}} - \frac{8 i}{n^{3}}$