Algebra Examples

${(\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} i)}^{2}$

Step 1

Simplify terms.

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Step 1.1

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

${(\frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2})}^{2}$

Step 1.2

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

$(\frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2}) (\frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2})$

Step 2

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

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Step 2.1

Apply the distributive property.

$\frac{\sqrt{2}}{2} (\frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2}) + \frac{\sqrt{2} i}{2} (\frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2})$

Step 2.2

Apply the distributive property.

$\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2} i}{2} + \frac{\sqrt{2} i}{2} (\frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2})$

Step 2.3

Apply the distributive property.

$\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2} i}{2} + \frac{\sqrt{2} i}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2} \cdot \frac{\sqrt{2} i}{2}$

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 $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2}}{2}$ .

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Step 3.1.1.1

Multiply $\frac{\sqrt{2}}{2}$ by $\frac{\sqrt{2}}{2}$ .

$\frac{\sqrt{2} \sqrt{2}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2} i}{2} + \frac{\sqrt{2} i}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2} \cdot \frac{\sqrt{2} i}{2}$

Step 3.1.1.2

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{{\sqrt{2}}^{1} \sqrt{2}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2} i}{2} + \frac{\sqrt{2} i}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2} \cdot \frac{\sqrt{2} i}{2}$

Step 3.1.1.3

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2} i}{2} + \frac{\sqrt{2} i}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2} \cdot \frac{\sqrt{2} i}{2}$

Step 3.1.1.4

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

$\frac{{\sqrt{2}}^{1 + 1}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2} i}{2} + \frac{\sqrt{2} i}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2} \cdot \frac{\sqrt{2} i}{2}$

Step 3.1.1.5

Add $1$ and $1$ .

$\frac{{\sqrt{2}}^{2}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2} i}{2} + \frac{\sqrt{2} i}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2} \cdot \frac{\sqrt{2} i}{2}$

Step 3.1.1.6

Multiply $2$ by $2$ .

$\frac{{\sqrt{2}}^{2}}{4} + \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2} i}{2} + \frac{\sqrt{2} i}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2} \cdot \frac{\sqrt{2} i}{2}$

Step 3.1.2

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 3.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

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

Step 3.1.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 3.1.2.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{2^{\frac{2}{2}}}{4} + \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2} i}{2} + \frac{\sqrt{2} i}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2} \cdot \frac{\sqrt{2} i}{2}$

Step 3.1.2.4

Cancel the common factor of $2$ .

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Step 3.1.2.4.1

Cancel the common factor.

$\frac{2^{\frac{2}{2}}}{4} + \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2} i}{2} + \frac{\sqrt{2} i}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2} \cdot \frac{\sqrt{2} i}{2}$

Step 3.1.2.4.2

Rewrite the expression.

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

Step 3.1.2.5

Evaluate the exponent.

$\frac{2}{4} + \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2} i}{2} + \frac{\sqrt{2} i}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2} \cdot \frac{\sqrt{2} i}{2}$

Step 3.1.3

Cancel the common factor of $2$ and $4$ .

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Step 3.1.3.1

Factor $2$ out of $2$ .

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

Step 3.1.3.2

Cancel the common factors.

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Step 3.1.3.2.1

Factor $2$ out of $4$ .

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

Step 3.1.3.2.2

Cancel the common factor.

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

Step 3.1.3.2.3

Rewrite the expression.

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

Step 3.1.4

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

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Step 3.1.4.1

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

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

Step 3.1.4.2

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{1}{2} + \frac{{\sqrt{2}}^{1} \sqrt{2} i}{2 \cdot 2} + \frac{\sqrt{2} i}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2} \cdot \frac{\sqrt{2} i}{2}$

Step 3.1.4.3

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{1}{2} + \frac{{\sqrt{2}}^{1} {\sqrt{2}}^{1} i}{2 \cdot 2} + \frac{\sqrt{2} i}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2} \cdot \frac{\sqrt{2} i}{2}$

Step 3.1.4.4

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

$\frac{1}{2} + \frac{{\sqrt{2}}^{1 + 1} i}{2 \cdot 2} + \frac{\sqrt{2} i}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2} \cdot \frac{\sqrt{2} i}{2}$

Step 3.1.4.5

Add $1$ and $1$ .

$\frac{1}{2} + \frac{{\sqrt{2}}^{2} i}{2 \cdot 2} + \frac{\sqrt{2} i}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2} \cdot \frac{\sqrt{2} i}{2}$

Step 3.1.4.6

Multiply $2$ by $2$ .

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

Step 3.1.5

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 3.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

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

Step 3.1.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 3.1.5.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 3.1.5.4

Cancel the common factor of $2$ .

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Step 3.1.5.4.1

Cancel the common factor.

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

Step 3.1.5.4.2

Rewrite the expression.

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

Step 3.1.5.5

Evaluate the exponent.

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

Step 3.1.6

Cancel the common factor of $2$ and $4$ .

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Step 3.1.6.1

Factor $2$ out of $2 i$ .

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

Step 3.1.6.2

Cancel the common factors.

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Step 3.1.6.2.1

Factor $2$ out of $4$ .

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

Step 3.1.6.2.2

Cancel the common factor.

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

Step 3.1.6.2.3

Rewrite the expression.

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

Step 3.1.7

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

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Step 3.1.7.1

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

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

Step 3.1.7.2

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{1}{2} + \frac{i}{2} + \frac{{\sqrt{2}}^{1} \sqrt{2} i}{2 \cdot 2} + \frac{\sqrt{2} i}{2} \cdot \frac{\sqrt{2} i}{2}$

Step 3.1.7.3

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{1}{2} + \frac{i}{2} + \frac{{\sqrt{2}}^{1} {\sqrt{2}}^{1} i}{2 \cdot 2} + \frac{\sqrt{2} i}{2} \cdot \frac{\sqrt{2} i}{2}$

Step 3.1.7.4

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

$\frac{1}{2} + \frac{i}{2} + \frac{{\sqrt{2}}^{1 + 1} i}{2 \cdot 2} + \frac{\sqrt{2} i}{2} \cdot \frac{\sqrt{2} i}{2}$

Step 3.1.7.5

Add $1$ and $1$ .

$\frac{1}{2} + \frac{i}{2} + \frac{{\sqrt{2}}^{2} i}{2 \cdot 2} + \frac{\sqrt{2} i}{2} \cdot \frac{\sqrt{2} i}{2}$

Step 3.1.7.6

Multiply $2$ by $2$ .

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

Step 3.1.8

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 3.1.8.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

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

Step 3.1.8.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 3.1.8.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 3.1.8.4

Cancel the common factor of $2$ .

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Step 3.1.8.4.1

Cancel the common factor.

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

Step 3.1.8.4.2

Rewrite the expression.

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

Step 3.1.8.5

Evaluate the exponent.

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

Step 3.1.9

Cancel the common factor of $2$ and $4$ .

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Step 3.1.9.1

Factor $2$ out of $2 i$ .

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

Step 3.1.9.2

Cancel the common factors.

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Step 3.1.9.2.1

Factor $2$ out of $4$ .

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

Step 3.1.9.2.2

Cancel the common factor.

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

Step 3.1.9.2.3

Rewrite the expression.

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

Step 3.1.10

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

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Step 3.1.10.1

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

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

Step 3.1.10.2

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{1}{2} + \frac{i}{2} + \frac{i}{2} + \frac{{\sqrt{2}}^{1} \sqrt{2} i i}{2 \cdot 2}$

Step 3.1.10.3

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{1}{2} + \frac{i}{2} + \frac{i}{2} + \frac{{\sqrt{2}}^{1} {\sqrt{2}}^{1} i i}{2 \cdot 2}$

Step 3.1.10.4

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

$\frac{1}{2} + \frac{i}{2} + \frac{i}{2} + \frac{{\sqrt{2}}^{1 + 1} i i}{2 \cdot 2}$

Step 3.1.10.5

Add $1$ and $1$ .

$\frac{1}{2} + \frac{i}{2} + \frac{i}{2} + \frac{{\sqrt{2}}^{2} i i}{2 \cdot 2}$

Step 3.1.10.6

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

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

Step 3.1.10.7

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

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

Step 3.1.10.8

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

$\frac{1}{2} + \frac{i}{2} + \frac{i}{2} + \frac{{\sqrt{2}}^{2} i^{1 + 1}}{2 \cdot 2}$

Step 3.1.10.9

Add $1$ and $1$ .

$\frac{1}{2} + \frac{i}{2} + \frac{i}{2} + \frac{{\sqrt{2}}^{2} i^{2}}{2 \cdot 2}$

Step 3.1.10.10

Multiply $2$ by $2$ .

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

Step 3.1.11

Simplify the numerator.

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Step 3.1.11.1

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 3.1.11.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

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

Step 3.1.11.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 3.1.11.1.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 3.1.11.1.4

Cancel the common factor of $2$ .

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Step 3.1.11.1.4.1

Cancel the common factor.

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

Step 3.1.11.1.4.2

Rewrite the expression.

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

Step 3.1.11.1.5

Evaluate the exponent.

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

Step 3.1.11.2

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

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

Step 3.1.12

Multiply $2$ by $- 1$ .

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

Step 3.1.13

Cancel the common factor of $- 2$ and $4$ .

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Step 3.1.13.1

Factor $2$ out of $- 2$ .

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

Step 3.1.13.2

Cancel the common factors.

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Step 3.1.13.2.1

Factor $2$ out of $4$ .

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

Step 3.1.13.2.2

Cancel the common factor.

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

Step 3.1.13.2.3

Rewrite the expression.

$\frac{1}{2} + \frac{i}{2} + \frac{i}{2} + \frac{- 1}{2}$

Step 3.1.14

Move the negative in front of the fraction.

$\frac{1}{2} + \frac{i}{2} + \frac{i}{2} - \frac{1}{2}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{1 - 1}{2} + \frac{i}{2} + \frac{i}{2}$

Step 3.3

Subtract $1$ from $1$ .

$\frac{0}{2} + \frac{i}{2} + \frac{i}{2}$

Step 3.4

Combine the numerators over the common denominator.

$\frac{0}{2} + \frac{2 i}{2}$

Step 4

Simplify each term.

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Step 4.1

Divide $0$ by $2$ .

$0 + \frac{2 i}{2}$

Step 4.2

Cancel the common factor of $2$ .

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Step 4.2.1

Cancel the common factor.

$0 + \frac{2 i}{2}$

Step 4.2.2

Divide $i$ by $1$ .

$0 + i$

Step 5

Add $0$ and $i$ .

$i$