Algebra Examples

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

Step 1

Simplify terms.

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Step 1.1

Combine $\frac{\sqrt{3}}{3}$ and $i$ .

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

Step 1.2

Rewrite ${(\frac{\sqrt{3}}{2} + \frac{\sqrt{3} i}{3})}^{2}$ as $(\frac{\sqrt{3}}{2} + \frac{\sqrt{3} i}{3}) (\frac{\sqrt{3}}{2} + \frac{\sqrt{3} i}{3})$ .

$(\frac{\sqrt{3}}{2} + \frac{\sqrt{3} i}{3}) (\frac{\sqrt{3}}{2} + \frac{\sqrt{3} i}{3})$

Step 2

Expand $(\frac{\sqrt{3}}{2} + \frac{\sqrt{3} i}{3}) (\frac{\sqrt{3}}{2} + \frac{\sqrt{3} i}{3})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$\frac{\sqrt{3}}{2} (\frac{\sqrt{3}}{2} + \frac{\sqrt{3} i}{3}) + \frac{\sqrt{3} i}{3} (\frac{\sqrt{3}}{2} + \frac{\sqrt{3} i}{3})$

Step 2.2

Apply the distributive property.

$\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3} i}{3} + \frac{\sqrt{3} i}{3} (\frac{\sqrt{3}}{2} + \frac{\sqrt{3} i}{3})$

Step 2.3

Apply the distributive property.

$\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3} i}{3} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3} i}{3}$

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 $\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}$ .

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Step 3.1.1.1

Multiply $\frac{\sqrt{3}}{2}$ by $\frac{\sqrt{3}}{2}$ .

$\frac{\sqrt{3} \sqrt{3}}{2 \cdot 2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3} i}{3} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3} i}{3}$

Step 3.1.1.2

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{{\sqrt{3}}^{1} \sqrt{3}}{2 \cdot 2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3} i}{3} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3} i}{3}$

Step 3.1.1.3

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}{2 \cdot 2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3} i}{3} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3} i}{3}$

Step 3.1.1.4

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

$\frac{{\sqrt{3}}^{1 + 1}}{2 \cdot 2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3} i}{3} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3} i}{3}$

Step 3.1.1.5

Add $1$ and $1$ .

$\frac{{\sqrt{3}}^{2}}{2 \cdot 2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3} i}{3} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3} i}{3}$

Step 3.1.1.6

Multiply $2$ by $2$ .

$\frac{{\sqrt{3}}^{2}}{4} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3} i}{3} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3} i}{3}$

Step 3.1.2

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 3.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{{(3^{\frac{1}{2}})}^{2}}{4} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3} i}{3} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3} i}{3}$

Step 3.1.2.2

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

$\frac{3^{\frac{1}{2} \cdot 2}}{4} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3} i}{3} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3} i}{3}$

Step 3.1.2.3

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

$\frac{3^{\frac{2}{2}}}{4} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3} i}{3} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3} i}{3}$

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.

$\frac{3^{\frac{2}{2}}}{4} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3} i}{3} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3} i}{3}$

Step 3.1.2.4.2

Rewrite the expression.

$\frac{3^{1}}{4} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3} i}{3} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3} i}{3}$

Step 3.1.2.5

Evaluate the exponent.

$\frac{3}{4} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3} i}{3} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3} i}{3}$

Step 3.1.3

Multiply $\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3} i}{3}$ .

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Step 3.1.3.1

Multiply $\frac{\sqrt{3}}{2}$ by $\frac{\sqrt{3} i}{3}$ .

$\frac{3}{4} + \frac{\sqrt{3} (\sqrt{3} i)}{2 \cdot 3} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3} i}{3}$

Step 3.1.3.2

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{3}{4} + \frac{{\sqrt{3}}^{1} \sqrt{3} i}{2 \cdot 3} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3} i}{3}$

Step 3.1.3.3

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{3}{4} + \frac{{\sqrt{3}}^{1} {\sqrt{3}}^{1} i}{2 \cdot 3} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3} i}{3}$

Step 3.1.3.4

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

$\frac{3}{4} + \frac{{\sqrt{3}}^{1 + 1} i}{2 \cdot 3} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3} i}{3}$

Step 3.1.3.5

Add $1$ and $1$ .

$\frac{3}{4} + \frac{{\sqrt{3}}^{2} i}{2 \cdot 3} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3} i}{3}$

Step 3.1.3.6

Multiply $2$ by $3$ .

$\frac{3}{4} + \frac{{\sqrt{3}}^{2} i}{6} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3} i}{3}$

Step 3.1.4

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 3.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

Step 3.1.4.2

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

$\frac{3}{4} + \frac{3^{\frac{1}{2} \cdot 2} i}{6} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3} i}{3}$

Step 3.1.4.3

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

$\frac{3}{4} + \frac{3^{\frac{2}{2}} i}{6} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3} i}{3}$

Step 3.1.4.4

Cancel the common factor of $2$ .

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Step 3.1.4.4.1

Cancel the common factor.

$\frac{3}{4} + \frac{3^{\frac{2}{2}} i}{6} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3} i}{3}$

Step 3.1.4.4.2

Rewrite the expression.

$\frac{3}{4} + \frac{3^{1} i}{6} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3} i}{3}$

Step 3.1.4.5

Evaluate the exponent.

$\frac{3}{4} + \frac{3 i}{6} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3} i}{3}$

Step 3.1.5

Cancel the common factor of $3$ and $6$ .

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Step 3.1.5.1

Factor $3$ out of $3 i$ .

$\frac{3}{4} + \frac{3 (i)}{6} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3} i}{3}$

Step 3.1.5.2

Cancel the common factors.

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Step 3.1.5.2.1

Factor $3$ out of $6$ .

$\frac{3}{4} + \frac{3 i}{3 \cdot 2} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3} i}{3}$

Step 3.1.5.2.2

Cancel the common factor.

$\frac{3}{4} + \frac{3 i}{3 \cdot 2} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3} i}{3}$

Step 3.1.5.2.3

Rewrite the expression.

$\frac{3}{4} + \frac{i}{2} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3} i}{3}$

Step 3.1.6

Multiply $\frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3}}{2}$ .

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Step 3.1.6.1

Multiply $\frac{\sqrt{3} i}{3}$ by $\frac{\sqrt{3}}{2}$ .

$\frac{3}{4} + \frac{i}{2} + \frac{\sqrt{3} i \sqrt{3}}{3 \cdot 2} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3} i}{3}$

Step 3.1.6.2

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{3}{4} + \frac{i}{2} + \frac{{\sqrt{3}}^{1} \sqrt{3} i}{3 \cdot 2} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3} i}{3}$

Step 3.1.6.3

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{3}{4} + \frac{i}{2} + \frac{{\sqrt{3}}^{1} {\sqrt{3}}^{1} i}{3 \cdot 2} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3} i}{3}$

Step 3.1.6.4

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

$\frac{3}{4} + \frac{i}{2} + \frac{{\sqrt{3}}^{1 + 1} i}{3 \cdot 2} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3} i}{3}$

Step 3.1.6.5

Add $1$ and $1$ .

$\frac{3}{4} + \frac{i}{2} + \frac{{\sqrt{3}}^{2} i}{3 \cdot 2} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3} i}{3}$

Step 3.1.6.6

Multiply $3$ by $2$ .

$\frac{3}{4} + \frac{i}{2} + \frac{{\sqrt{3}}^{2} i}{6} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3} i}{3}$

Step 3.1.7

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 3.1.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

Step 3.1.7.2

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

$\frac{3}{4} + \frac{i}{2} + \frac{3^{\frac{1}{2} \cdot 2} i}{6} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3} i}{3}$

Step 3.1.7.3

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

$\frac{3}{4} + \frac{i}{2} + \frac{3^{\frac{2}{2}} i}{6} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3} i}{3}$

Step 3.1.7.4

Cancel the common factor of $2$ .

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Step 3.1.7.4.1

Cancel the common factor.

$\frac{3}{4} + \frac{i}{2} + \frac{3^{\frac{2}{2}} i}{6} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3} i}{3}$

Step 3.1.7.4.2

Rewrite the expression.

$\frac{3}{4} + \frac{i}{2} + \frac{3^{1} i}{6} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3} i}{3}$

Step 3.1.7.5

Evaluate the exponent.

$\frac{3}{4} + \frac{i}{2} + \frac{3 i}{6} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3} i}{3}$

Step 3.1.8

Cancel the common factor of $3$ and $6$ .

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Step 3.1.8.1

Factor $3$ out of $3 i$ .

$\frac{3}{4} + \frac{i}{2} + \frac{3 (i)}{6} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3} i}{3}$

Step 3.1.8.2

Cancel the common factors.

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Step 3.1.8.2.1

Factor $3$ out of $6$ .

$\frac{3}{4} + \frac{i}{2} + \frac{3 i}{3 \cdot 2} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3} i}{3}$

Step 3.1.8.2.2

Cancel the common factor.

$\frac{3}{4} + \frac{i}{2} + \frac{3 i}{3 \cdot 2} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3} i}{3}$

Step 3.1.8.2.3

Rewrite the expression.

$\frac{3}{4} + \frac{i}{2} + \frac{i}{2} + \frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3} i}{3}$

Step 3.1.9

Multiply $\frac{\sqrt{3} i}{3} \cdot \frac{\sqrt{3} i}{3}$ .

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Step 3.1.9.1

Multiply $\frac{\sqrt{3} i}{3}$ by $\frac{\sqrt{3} i}{3}$ .

$\frac{3}{4} + \frac{i}{2} + \frac{i}{2} + \frac{\sqrt{3} i (\sqrt{3} i)}{3 \cdot 3}$

Step 3.1.9.2

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{3}{4} + \frac{i}{2} + \frac{i}{2} + \frac{{\sqrt{3}}^{1} \sqrt{3} i i}{3 \cdot 3}$

Step 3.1.9.3

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{3}{4} + \frac{i}{2} + \frac{i}{2} + \frac{{\sqrt{3}}^{1} {\sqrt{3}}^{1} i i}{3 \cdot 3}$

Step 3.1.9.4

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

$\frac{3}{4} + \frac{i}{2} + \frac{i}{2} + \frac{{\sqrt{3}}^{1 + 1} i i}{3 \cdot 3}$

Step 3.1.9.5

Add $1$ and $1$ .

$\frac{3}{4} + \frac{i}{2} + \frac{i}{2} + \frac{{\sqrt{3}}^{2} i i}{3 \cdot 3}$

Step 3.1.9.6

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

$\frac{3}{4} + \frac{i}{2} + \frac{i}{2} + \frac{{\sqrt{3}}^{2} (i^{1} i)}{3 \cdot 3}$

Step 3.1.9.7

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

$\frac{3}{4} + \frac{i}{2} + \frac{i}{2} + \frac{{\sqrt{3}}^{2} (i^{1} i^{1})}{3 \cdot 3}$

Step 3.1.9.8

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

$\frac{3}{4} + \frac{i}{2} + \frac{i}{2} + \frac{{\sqrt{3}}^{2} i^{1 + 1}}{3 \cdot 3}$

Step 3.1.9.9

Add $1$ and $1$ .

$\frac{3}{4} + \frac{i}{2} + \frac{i}{2} + \frac{{\sqrt{3}}^{2} i^{2}}{3 \cdot 3}$

Step 3.1.9.10

Multiply $3$ by $3$ .

$\frac{3}{4} + \frac{i}{2} + \frac{i}{2} + \frac{{\sqrt{3}}^{2} i^{2}}{9}$

Step 3.1.10

Simplify the numerator.

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Step 3.1.10.1

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 3.1.10.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{3}{4} + \frac{i}{2} + \frac{i}{2} + \frac{{(3^{\frac{1}{2}})}^{2} i^{2}}{9}$

Step 3.1.10.1.2

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

$\frac{3}{4} + \frac{i}{2} + \frac{i}{2} + \frac{3^{\frac{1}{2} \cdot 2} i^{2}}{9}$

Step 3.1.10.1.3

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

$\frac{3}{4} + \frac{i}{2} + \frac{i}{2} + \frac{3^{\frac{2}{2}} i^{2}}{9}$

Step 3.1.10.1.4

Cancel the common factor of $2$ .

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Step 3.1.10.1.4.1

Cancel the common factor.

$\frac{3}{4} + \frac{i}{2} + \frac{i}{2} + \frac{3^{\frac{2}{2}} i^{2}}{9}$

Step 3.1.10.1.4.2

Rewrite the expression.

$\frac{3}{4} + \frac{i}{2} + \frac{i}{2} + \frac{3^{1} i^{2}}{9}$

Step 3.1.10.1.5

Evaluate the exponent.

$\frac{3}{4} + \frac{i}{2} + \frac{i}{2} + \frac{3 i^{2}}{9}$

Step 3.1.10.2

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

$\frac{3}{4} + \frac{i}{2} + \frac{i}{2} + \frac{3 \cdot - 1}{9}$

Step 3.1.11

Multiply $3$ by $- 1$ .

$\frac{3}{4} + \frac{i}{2} + \frac{i}{2} + \frac{- 3}{9}$

Step 3.1.12

Cancel the common factor of $- 3$ and $9$ .

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Step 3.1.12.1

Factor $3$ out of $- 3$ .

$\frac{3}{4} + \frac{i}{2} + \frac{i}{2} + \frac{3 (- 1)}{9}$

Step 3.1.12.2

Cancel the common factors.

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Step 3.1.12.2.1

Factor $3$ out of $9$ .

$\frac{3}{4} + \frac{i}{2} + \frac{i}{2} + \frac{3 \cdot - 1}{3 \cdot 3}$

Step 3.1.12.2.2

Cancel the common factor.

$\frac{3}{4} + \frac{i}{2} + \frac{i}{2} + \frac{3 \cdot - 1}{3 \cdot 3}$

Step 3.1.12.2.3

Rewrite the expression.

$\frac{3}{4} + \frac{i}{2} + \frac{i}{2} + \frac{- 1}{3}$

Step 3.1.13

Move the negative in front of the fraction.

$\frac{3}{4} + \frac{i}{2} + \frac{i}{2} - \frac{1}{3}$

Step 3.2

To write $\frac{3}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{3}{4} \cdot \frac{3}{3} - \frac{1}{3} + \frac{i}{2} + \frac{i}{2}$

Step 3.3

To write $- \frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{3}{4} \cdot \frac{3}{3} - \frac{1}{3} \cdot \frac{4}{4} + \frac{i}{2} + \frac{i}{2}$

Step 3.4

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.1

Multiply $\frac{3}{4}$ by $\frac{3}{3}$ .

$\frac{3 \cdot 3}{4 \cdot 3} - \frac{1}{3} \cdot \frac{4}{4} + \frac{i}{2} + \frac{i}{2}$

Step 3.4.2

Multiply $4$ by $3$ .

$\frac{3 \cdot 3}{12} - \frac{1}{3} \cdot \frac{4}{4} + \frac{i}{2} + \frac{i}{2}$

Step 3.4.3

Multiply $\frac{1}{3}$ by $\frac{4}{4}$ .

$\frac{3 \cdot 3}{12} - \frac{4}{3 \cdot 4} + \frac{i}{2} + \frac{i}{2}$

Step 3.4.4

Multiply $3$ by $4$ .

$\frac{3 \cdot 3}{12} - \frac{4}{12} + \frac{i}{2} + \frac{i}{2}$

Step 3.5

Combine the numerators over the common denominator.

$\frac{3 \cdot 3 - 4}{12} + \frac{i}{2} + \frac{i}{2}$

Step 3.6

Simplify the numerator.

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Step 3.6.1

Multiply $3$ by $3$ .

$\frac{9 - 4}{12} + \frac{i}{2} + \frac{i}{2}$

Step 3.6.2

Subtract $4$ from $9$ .

$\frac{5}{12} + \frac{i}{2} + \frac{i}{2}$

Step 3.7

Combine the numerators over the common denominator.

$\frac{5}{12} + \frac{2 i}{2}$

Step 4

Cancel the common factor of $2$ .

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Step 4.1

Cancel the common factor.

$\frac{5}{12} + \frac{2 i}{2}$

Step 4.2

Divide $i$ by $1$ .

$\frac{5}{12} + i$