Algebra Examples

${(\cos (\frac{π}{4}) + i \sin (\frac{π}{4}))}^{3}$

Step 1

Simplify each term.

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Step 1.1

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

${(\frac{\sqrt{2}}{2} + i \sin (\frac{π}{4}))}^{3}$

Step 1.2

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

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

Step 1.3

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

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

Step 2

Use the Binomial Theorem.

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

Step 3

Simplify terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

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

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

Step 3.1.2

Simplify the numerator.

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Step 3.1.2.1

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

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

Step 3.1.2.2

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

$\frac{\sqrt{8}}{2^{3}} + 3 {(\frac{\sqrt{2}}{2})}^{2} \frac{i \sqrt{2}}{2} + 3 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{2} + {(\frac{i \sqrt{2}}{2})}^{3}$

Step 3.1.2.3

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 3.1.2.3.1

Factor $4$ out of $8$ .

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

Step 3.1.2.3.2

Rewrite $4$ as $2^{2}$ .

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

Step 3.1.2.4

Pull terms out from under the radical.

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

Step 3.1.3

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

$\frac{2 \sqrt{2}}{8} + 3 {(\frac{\sqrt{2}}{2})}^{2} \frac{i \sqrt{2}}{2} + 3 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{2} + {(\frac{i \sqrt{2}}{2})}^{3}$

Step 3.1.4

Cancel the common factor of $2$ and $8$ .

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Step 3.1.4.1

Factor $2$ out of $2 \sqrt{2}$ .

$\frac{2 (\sqrt{2})}{8} + 3 {(\frac{\sqrt{2}}{2})}^{2} \frac{i \sqrt{2}}{2} + 3 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{2} + {(\frac{i \sqrt{2}}{2})}^{3}$

Step 3.1.4.2

Cancel the common factors.

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Step 3.1.4.2.1

Factor $2$ out of $8$ .

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

Step 3.1.4.2.2

Cancel the common factor.

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

Step 3.1.4.2.3

Rewrite the expression.

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

Step 3.1.5

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

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

Step 3.1.6

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

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Step 3.1.6.1

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

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

Step 3.1.6.2

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

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

Step 3.1.6.3

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

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

Step 3.1.6.4

Cancel the common factor of $2$ .

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Step 3.1.6.4.1

Cancel the common factor.

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

Step 3.1.6.4.2

Rewrite the expression.

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

Step 3.1.6.5

Evaluate the exponent.

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

Step 3.1.7

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

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

Step 3.1.8

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

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Step 3.1.8.1

Factor $2$ out of $2$ .

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

Step 3.1.8.2

Cancel the common factors.

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Step 3.1.8.2.1

Factor $2$ out of $4$ .

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

Step 3.1.8.2.2

Cancel the common factor.

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

Step 3.1.8.2.3

Rewrite the expression.

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

Step 3.1.9

Combine $3$ and $\frac{1}{2}$ .

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

Step 3.1.10

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

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Step 3.1.10.1

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

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

Step 3.1.10.2

Multiply $2$ by $2$ .

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

Step 3.1.11

Combine $3$ and $\frac{\sqrt{2}}{2}$ .

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

Step 3.1.12

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

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Step 3.1.12.1

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

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

Step 3.1.12.2

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

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

Step 3.1.13

Combine.

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

Step 3.1.14

Multiply $\sqrt{2}$ by ${\sqrt{2}}^{2}$ by adding the exponents.

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Step 3.1.14.1

Move ${\sqrt{2}}^{2}$ .

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

Step 3.1.14.2

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

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Step 3.1.14.2.1

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

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

Step 3.1.14.2.2

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

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

Step 3.1.14.3

Add $2$ and $1$ .

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

Step 3.1.15

Multiply $2$ by $2^{2}$ by adding the exponents.

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Step 3.1.15.1

Multiply $2$ by $2^{2}$ .

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Step 3.1.15.1.1

Raise $2$ to the power of $1$ .

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

Step 3.1.15.1.2

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

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

Step 3.1.15.2

Add $1$ and $2$ .

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

Step 3.1.16

Simplify the numerator.

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Step 3.1.16.1

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

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

Step 3.1.16.2

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

$\frac{\sqrt{2}}{4} + \frac{3 i \sqrt{2}}{4} + \frac{3 \sqrt{8} i^{2}}{2^{3}} + {(\frac{i \sqrt{2}}{2})}^{3}$

Step 3.1.16.3

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 3.1.16.3.1

Factor $4$ out of $8$ .

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

Step 3.1.16.3.2

Rewrite $4$ as $2^{2}$ .

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

Step 3.1.16.4

Pull terms out from under the radical.

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

Step 3.1.16.5

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

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

Step 3.1.16.6

Combine exponents.

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Step 3.1.16.6.1

Multiply $3$ by $2$ .

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

Step 3.1.16.6.2

Multiply $- 1$ by $6$ .

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

Step 3.1.17

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

$\frac{\sqrt{2}}{4} + \frac{3 i \sqrt{2}}{4} + \frac{- 6 \sqrt{2}}{8} + {(\frac{i \sqrt{2}}{2})}^{3}$

Step 3.1.18

Cancel the common factor of $- 6$ and $8$ .

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Step 3.1.18.1

Factor $2$ out of $- 6 \sqrt{2}$ .

$\frac{\sqrt{2}}{4} + \frac{3 i \sqrt{2}}{4} + \frac{2 (- 3 \sqrt{2})}{8} + {(\frac{i \sqrt{2}}{2})}^{3}$

Step 3.1.18.2

Cancel the common factors.

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Step 3.1.18.2.1

Factor $2$ out of $8$ .

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

Step 3.1.18.2.2

Cancel the common factor.

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

Step 3.1.18.2.3

Rewrite the expression.

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

Step 3.1.19

Move the negative in front of the fraction.

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

Step 3.1.20

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

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Step 3.1.20.1

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

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

Step 3.1.20.2

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

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

Step 3.1.21

Simplify the numerator.

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Step 3.1.21.1

Factor out $i^{2}$ .

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

Step 3.1.21.2

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

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

Step 3.1.21.3

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

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

Step 3.1.21.4

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

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

Step 3.1.21.5

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

$\frac{\sqrt{2}}{4} + \frac{3 i \sqrt{2}}{4} - \frac{3 \sqrt{2}}{4} + \frac{- i \sqrt{8}}{2^{3}}$

Step 3.1.21.6

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 3.1.21.6.1

Factor $4$ out of $8$ .

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

Step 3.1.21.6.2

Rewrite $4$ as $2^{2}$ .

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

Step 3.1.21.7

Pull terms out from under the radical.

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

Step 3.1.21.8

Multiply $2$ by $- 1$ .

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

Step 3.1.22

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

$\frac{\sqrt{2}}{4} + \frac{3 i \sqrt{2}}{4} - \frac{3 \sqrt{2}}{4} + \frac{- 2 i \sqrt{2}}{8}$

Step 3.1.23

Cancel the common factor of $- 2$ and $8$ .

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Step 3.1.23.1

Factor $2$ out of $- 2 i \sqrt{2}$ .

$\frac{\sqrt{2}}{4} + \frac{3 i \sqrt{2}}{4} - \frac{3 \sqrt{2}}{4} + \frac{2 (- i \sqrt{2})}{8}$

Step 3.1.23.2

Cancel the common factors.

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Step 3.1.23.2.1

Factor $2$ out of $8$ .

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

Step 3.1.23.2.2

Cancel the common factor.

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

Step 3.1.23.2.3

Rewrite the expression.

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

Step 3.1.24

Move the negative in front of the fraction.

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

Step 3.2

Simplify terms.

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Step 3.2.1

Combine the numerators over the common denominator.

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

Step 3.2.2

Subtract $3 \sqrt{2}$ from $\sqrt{2}$ .

$\frac{3 i \sqrt{2} - 2 \sqrt{2} - i \sqrt{2}}{4}$

Step 3.2.3

Subtract $i \sqrt{2}$ from $3 i \sqrt{2}$ .

$\frac{2 i \sqrt{2} - 2 \sqrt{2}}{4}$

Step 3.2.4

Reorder $2 i \sqrt{2}$ and $- 2 \sqrt{2}$ .

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

Step 3.2.5

Factor $- 1$ out of $- 2 \sqrt{2}$ .

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

Step 3.2.6

Factor $- 1$ out of $2 i \sqrt{2}$ .

$\frac{- (2 \sqrt{2}) - (- 2 i \sqrt{2})}{4}$

Step 3.2.7

Factor $- 1$ out of $- (2 \sqrt{2}) - (- 2 i \sqrt{2})$ .

$\frac{- (2 \sqrt{2} - 2 i \sqrt{2})}{4}$

Step 3.2.8

Simplify the expression.

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Step 3.2.8.1

Rewrite $- (2 \sqrt{2} - 2 i \sqrt{2})$ as $- 1 (2 \sqrt{2} - 2 i \sqrt{2})$ .

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

Step 3.2.8.2

Move the negative in front of the fraction.

$- \frac{2 \sqrt{2} - 2 i \sqrt{2}}{4}$

Step 4

This is the trigonometric form of a complex number where $| z |$ is the modulus and $θ$ is the angle created on the complex plane.

$z = a + b i = | z | (\cos (θ) + i \sin (θ))$

Step 5

The modulus of a complex number is the distance from the origin on the complex plane.

$| z | = \sqrt{a^{2} + b^{2}}$ where $z = a + b i$

Step 6

Substitute the actual values of $a = 0$ and $b = - \frac{1}{4}$ .

$| z | = \sqrt{{(- \frac{1}{4})}^{2}}$

Step 7

Find $| z |$ .

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Step 7.1

Apply the product rule to $- \frac{1}{4}$ .

$| z | = \sqrt{{(- 1)}^{2} {(\frac{1}{4})}^{2}}$

Step 7.2

Raise $- 1$ to the power of $2$ .

$| z | = \sqrt{1 {(\frac{1}{4})}^{2}}$

Step 7.3

Apply the product rule to $\frac{1}{4}$ .

$| z | = \sqrt{1 (\frac{1^{2}}{4^{2}})}$

Step 7.4

One to any power is one.

$| z | = \sqrt{1 (\frac{1}{4^{2}})}$

Step 7.5

Raise $4$ to the power of $2$ .

$| z | = \sqrt{1 (\frac{1}{16})}$

Step 7.6

Multiply $\frac{1}{16}$ by $1$ .

$| z | = \sqrt{\frac{1}{16}}$

Step 7.7

Rewrite $\sqrt{\frac{1}{16}}$ as $\frac{\sqrt{1}}{\sqrt{16}}$ .

$| z | = \frac{\sqrt{1}}{\sqrt{16}}$

Step 7.8

Any root of $1$ is $1$ .

$| z | = \frac{1}{\sqrt{16}}$

Step 7.9

Simplify the denominator.

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Step 7.9.1

Rewrite $16$ as $4^{2}$ .

$| z | = \frac{1}{\sqrt{4^{2}}}$

Step 7.9.2

Pull terms out from under the radical, assuming positive real numbers.

$| z | = \frac{1}{4}$

Step 8

The angle of the point on the complex plane is the inverse tangent of the complex portion over the real portion.

$θ = \arctan (\frac{- \frac{1}{4}}{0})$

Step 9

Since the argument is undefined and $b$ is negative, the angle of the point on the complex plane is $\frac{3 π}{2}$ .

$θ = \frac{3 π}{2}$

Step 10

Substitute the values of $θ = \frac{3 π}{2}$ and $| z | = \frac{1}{4}$ .

$\frac{1}{4 (\cos (\frac{3 π}{2}) + i \sin (\frac{3 π}{2}))}$