Algebra Examples

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

Step 1

Rewrite ${(1 + \frac{2 i}{n})}^{2}$ as $(1 + \frac{2 i}{n}) (1 + \frac{2 i}{n})$ .

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

Step 2

Expand $(1 + \frac{2 i}{n}) (1 + \frac{2 i}{n})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

$1 \cdot 1 + 1 \frac{2 i}{n} + \frac{2 i}{n} \cdot 1 + \frac{2 i}{n} \cdot \frac{2 i}{n}$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $1$ by $1$ .

$1 + 1 \frac{2 i}{n} + \frac{2 i}{n} \cdot 1 + \frac{2 i}{n} \cdot \frac{2 i}{n}$

Step 3.1.2

Multiply $\frac{2 i}{n}$ by $1$ .

$1 + \frac{2 i}{n} + \frac{2 i}{n} \cdot 1 + \frac{2 i}{n} \cdot \frac{2 i}{n}$

Step 3.1.3

Multiply $\frac{2 i}{n}$ by $1$ .

$1 + \frac{2 i}{n} + \frac{2 i}{n} + \frac{2 i}{n} \cdot \frac{2 i}{n}$

Step 3.1.4

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

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Step 3.1.4.1

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

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

Step 3.1.4.2

Multiply $2$ by $2$ .

$1 + \frac{2 i}{n} + \frac{2 i}{n} + \frac{4 i i}{n \cdot n}$

Step 3.1.4.3

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

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

Step 3.1.4.4

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

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

Step 3.1.4.5

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

$1 + \frac{2 i}{n} + \frac{2 i}{n} + \frac{4 i^{1 + 1}}{n \cdot n}$

Step 3.1.4.6

Add $1$ and $1$ .

$1 + \frac{2 i}{n} + \frac{2 i}{n} + \frac{4 i^{2}}{n \cdot n}$

Step 3.1.4.7

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

$1 + \frac{2 i}{n} + \frac{2 i}{n} + \frac{4 i^{2}}{n^{1} n}$

Step 3.1.4.8

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

$1 + \frac{2 i}{n} + \frac{2 i}{n} + \frac{4 i^{2}}{n^{1} n^{1}}$

Step 3.1.4.9

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

$1 + \frac{2 i}{n} + \frac{2 i}{n} + \frac{4 i^{2}}{n^{1 + 1}}$

Step 3.1.4.10

Add $1$ and $1$ .

$1 + \frac{2 i}{n} + \frac{2 i}{n} + \frac{4 i^{2}}{n^{2}}$

Step 3.1.5

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

$1 + \frac{2 i}{n} + \frac{2 i}{n} + \frac{4 \cdot - 1}{n^{2}}$

Step 3.1.6

Multiply $4$ by $- 1$ .

$1 + \frac{2 i}{n} + \frac{2 i}{n} + \frac{- 4}{n^{2}}$

Step 3.1.7

Move the negative in front of the fraction.

$1 + \frac{2 i}{n} + \frac{2 i}{n} - \frac{4}{n^{2}}$

Step 3.2

Add $\frac{2 i}{n}$ and $\frac{2 i}{n}$ .

$1 + 2 \frac{2 i}{n} - \frac{4}{n^{2}}$

Step 4

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

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Step 4.1

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

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

Step 4.2

Multiply $2$ by $2$ .

$1 + \frac{4 i}{n} - \frac{4}{n^{2}}$