Calculus Examples

${(\cos (\frac{3 π}{12}) + i \sin (\frac{3 π}{12}))}^{5}$

Step 1

Simplify each term.

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Step 1.1

Cancel the common factor of $3$ and $12$ .

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Step 1.1.1

Factor $3$ out of $3 π$ .

${(\cos (\frac{3 (π)}{12}) + i \sin (\frac{3 π}{12}))}^{5}$

Step 1.1.2

Cancel the common factors.

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Step 1.1.2.1

Factor $3$ out of $12$ .

${(\cos (\frac{3 π}{3 \cdot 4}) + i \sin (\frac{3 π}{12}))}^{5}$

Step 1.1.2.2

Cancel the common factor.

${(\cos (\frac{3 π}{3 \cdot 4}) + i \sin (\frac{3 π}{12}))}^{5}$

Step 1.1.2.3

Rewrite the expression.

${(\cos (\frac{π}{4}) + i \sin (\frac{3 π}{12}))}^{5}$

Step 1.2

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

${(\frac{\sqrt{2}}{2} + i \sin (\frac{3 π}{12}))}^{5}$

Step 1.3

Cancel the common factor of $3$ and $12$ .

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Step 1.3.1

Factor $3$ out of $3 π$ .

${(\frac{\sqrt{2}}{2} + i \sin (\frac{3 (π)}{12}))}^{5}$

Step 1.3.2

Cancel the common factors.

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Step 1.3.2.1

Factor $3$ out of $12$ .

${(\frac{\sqrt{2}}{2} + i \sin (\frac{3 π}{3 \cdot 4}))}^{5}$

Step 1.3.2.2

Cancel the common factor.

${(\frac{\sqrt{2}}{2} + i \sin (\frac{3 π}{3 \cdot 4}))}^{5}$

Step 1.3.2.3

Rewrite the expression.

${(\frac{\sqrt{2}}{2} + i \sin (\frac{π}{4}))}^{5}$

Step 1.4

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

${(\frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2})}^{5}$

Step 1.5

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

${(\frac{\sqrt{2}}{2} + \frac{i \sqrt{2}}{2})}^{5}$

Step 2

Use the Binomial Theorem.

${(\frac{\sqrt{2}}{2})}^{5} + 5 {(\frac{\sqrt{2}}{2})}^{4} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

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

$\frac{{\sqrt{2}}^{5}}{2^{5}} + 5 {(\frac{\sqrt{2}}{2})}^{4} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.2

Simplify the numerator.

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Step 3.1.2.1

Rewrite ${\sqrt{2}}^{5}$ as $\sqrt{2^{5}}$ .

$\frac{\sqrt{2^{5}}}{2^{5}} + 5 {(\frac{\sqrt{2}}{2})}^{4} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.2.2

Raise $2$ to the power of $5$ .

$\frac{\sqrt{32}}{2^{5}} + 5 {(\frac{\sqrt{2}}{2})}^{4} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.2.3

Rewrite $32$ as $4^{2} \cdot 2$ .

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Step 3.1.2.3.1

Factor $16$ out of $32$ .

$\frac{\sqrt{16 (2)}}{2^{5}} + 5 {(\frac{\sqrt{2}}{2})}^{4} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.2.3.2

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

$\frac{\sqrt{4^{2} \cdot 2}}{2^{5}} + 5 {(\frac{\sqrt{2}}{2})}^{4} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.2.4

Pull terms out from under the radical.

$\frac{4 \sqrt{2}}{2^{5}} + 5 {(\frac{\sqrt{2}}{2})}^{4} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.3

Raise $2$ to the power of $5$ .

$\frac{4 \sqrt{2}}{32} + 5 {(\frac{\sqrt{2}}{2})}^{4} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.4

Cancel the common factor of $4$ and $32$ .

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Step 3.1.4.1

Factor $4$ out of $4 \sqrt{2}$ .

$\frac{4 (\sqrt{2})}{32} + 5 {(\frac{\sqrt{2}}{2})}^{4} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.4.2

Cancel the common factors.

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Step 3.1.4.2.1

Factor $4$ out of $32$ .

$\frac{4 \sqrt{2}}{4 \cdot 8} + 5 {(\frac{\sqrt{2}}{2})}^{4} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.4.2.2

Cancel the common factor.

Step 3.1.4.2.3

Rewrite the expression.

$\frac{\sqrt{2}}{8} + 5 {(\frac{\sqrt{2}}{2})}^{4} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.5

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

$\frac{\sqrt{2}}{8} + 5 \frac{{\sqrt{2}}^{4}}{2^{4}} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.6

Simplify the numerator.

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Step 3.1.6.1

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

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Step 3.1.6.1.1

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

$\frac{\sqrt{2}}{8} + 5 \frac{{(2^{\frac{1}{2}})}^{4}}{2^{4}} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.6.1.2

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

$\frac{\sqrt{2}}{8} + 5 \frac{2^{\frac{1}{2} \cdot 4}}{2^{4}} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.6.1.3

Combine $\frac{1}{2}$ and $4$ .

$\frac{\sqrt{2}}{8} + 5 \frac{2^{\frac{4}{2}}}{2^{4}} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.6.1.4

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

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Step 3.1.6.1.4.1

Factor $2$ out of $4$ .

$\frac{\sqrt{2}}{8} + 5 \frac{2^{\frac{2 \cdot 2}{2}}}{2^{4}} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.6.1.4.2

Cancel the common factors.

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Step 3.1.6.1.4.2.1

Factor $2$ out of $2$ .

$\frac{\sqrt{2}}{8} + 5 \frac{2^{\frac{2 \cdot 2}{2 (1)}}}{2^{4}} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.6.1.4.2.2

Cancel the common factor.

$\frac{\sqrt{2}}{8} + 5 \frac{2^{\frac{2 \cdot 2}{2 \cdot 1}}}{2^{4}} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.6.1.4.2.3

Rewrite the expression.

$\frac{\sqrt{2}}{8} + 5 \frac{2^{\frac{2}{1}}}{2^{4}} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.6.1.4.2.4

Divide $2$ by $1$ .

$\frac{\sqrt{2}}{8} + 5 \frac{2^{2}}{2^{4}} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.6.2

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

$\frac{\sqrt{2}}{8} + 5 \frac{4}{2^{4}} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.7

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

$\frac{\sqrt{2}}{8} + 5 (\frac{4}{16}) \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.8

Cancel the common factor of $4$ and $16$ .

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Step 3.1.8.1

Factor $4$ out of $4$ .

$\frac{\sqrt{2}}{8} + 5 \frac{4 (1)}{16} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.8.2

Cancel the common factors.

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Step 3.1.8.2.1

Factor $4$ out of $16$ .

$\frac{\sqrt{2}}{8} + 5 \frac{4 \cdot 1}{4 \cdot 4} \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.8.2.2

Cancel the common factor.

Step 3.1.8.2.3

Rewrite the expression.

$\frac{\sqrt{2}}{8} + 5 (\frac{1}{4}) \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.9

Combine $5$ and $\frac{1}{4}$ .

$\frac{\sqrt{2}}{8} + \frac{5}{4} \cdot \frac{i \sqrt{2}}{2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.10

Multiply $\frac{5}{4} \cdot \frac{i \sqrt{2}}{2}$ .

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Step 3.1.10.1

Multiply $\frac{5}{4}$ by $\frac{i \sqrt{2}}{2}$ .

$\frac{\sqrt{2}}{8} + \frac{5 (i \sqrt{2})}{4 \cdot 2} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.10.2

Multiply $4$ by $2$ .

$\frac{\sqrt{2}}{8} + \frac{5 (i \sqrt{2})}{8} + 10 {(\frac{\sqrt{2}}{2})}^{3} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.11

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

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + 10 \frac{{\sqrt{2}}^{3}}{2^{3}} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.12

Simplify the numerator.

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Step 3.1.12.1

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

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + 10 \frac{\sqrt{2^{3}}}{2^{3}} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.12.2

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

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + 10 \frac{\sqrt{8}}{2^{3}} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.12.3

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

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Step 3.1.12.3.1

Factor $4$ out of $8$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + 10 \frac{\sqrt{4 (2)}}{2^{3}} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.12.3.2

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

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + 10 \frac{\sqrt{2^{2} \cdot 2}}{2^{3}} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.12.4

Pull terms out from under the radical.

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + 10 \frac{2 \sqrt{2}}{2^{3}} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.13

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

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + 10 \frac{2 \sqrt{2}}{8} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.14

Cancel the common factor of $2$ .

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Step 3.1.14.1

Factor $2$ out of $10$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + 2 (5) \frac{2 \sqrt{2}}{8} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.14.2

Factor $2$ out of $8$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + 2 \cdot 5 \frac{2 \sqrt{2}}{2 \cdot 4} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.14.3

Cancel the common factor.

Step 3.1.14.4

Rewrite the expression.

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + 5 \frac{2 \sqrt{2}}{4} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.15

Combine $5$ and $\frac{2 \sqrt{2}}{4}$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{5 (2 \sqrt{2})}{4} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.16

Multiply $2$ by $5$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{10 \sqrt{2}}{4} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.17

Cancel the common factor of $10$ and $4$ .

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Step 3.1.17.1

Factor $2$ out of $10 \sqrt{2}$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{2 (5 \sqrt{2})}{4} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.17.2

Cancel the common factors.

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Step 3.1.17.2.1

Factor $2$ out of $4$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{2 (5 \sqrt{2})}{2 (2)} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.17.2.2

Cancel the common factor.

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{2 (5 \sqrt{2})}{2 \cdot 2} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.17.2.3

Rewrite the expression.

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{5 \sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{2} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.18

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

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Step 3.1.18.1

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

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{5 \sqrt{2}}{2} \cdot \frac{{(i \sqrt{2})}^{2}}{2^{2}} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.18.2

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

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{5 \sqrt{2}}{2} \cdot \frac{i^{2} {\sqrt{2}}^{2}}{2^{2}} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.19

Combine.

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{5 \sqrt{2} (i^{2} {\sqrt{2}}^{2})}{2 \cdot 2^{2}} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.20

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

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Step 3.1.20.1

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

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{5 ({\sqrt{2}}^{2} \sqrt{2}) i^{2}}{2 \cdot 2^{2}} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.20.2

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

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Step 3.1.20.2.1

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

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{5 ({\sqrt{2}}^{2} {\sqrt{2}}^{1}) i^{2}}{2 \cdot 2^{2}} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.20.2.2

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

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{5 {\sqrt{2}}^{2 + 1} i^{2}}{2 \cdot 2^{2}} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.20.3

Add $2$ and $1$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{5 {\sqrt{2}}^{3} i^{2}}{2 \cdot 2^{2}} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.21

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

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Step 3.1.21.1

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

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Step 3.1.21.1.1

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

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{5 {\sqrt{2}}^{3} i^{2}}{2^{1} \cdot 2^{2}} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.21.1.2

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

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{5 {\sqrt{2}}^{3} i^{2}}{2^{1 + 2}} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.21.2

Add $1$ and $2$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{5 {\sqrt{2}}^{3} i^{2}}{2^{3}} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.22

Simplify the numerator.

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Step 3.1.22.1

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

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{5 \sqrt{2^{3}} i^{2}}{2^{3}} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.22.2

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

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{5 \sqrt{8} i^{2}}{2^{3}} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.22.3

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

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Step 3.1.22.3.1

Factor $4$ out of $8$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{5 \sqrt{4 (2)} i^{2}}{2^{3}} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.22.3.2

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

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{5 \sqrt{2^{2} \cdot 2} i^{2}}{2^{3}} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.22.4

Pull terms out from under the radical.

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{5 (2 \sqrt{2}) i^{2}}{2^{3}} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.22.5

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

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{5 \cdot 2 \sqrt{2} \cdot - 1}{2^{3}} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.22.6

Combine exponents.

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Step 3.1.22.6.1

Multiply $5$ by $2$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{10 \sqrt{2} \cdot - 1}{2^{3}} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.22.6.2

Multiply $- 1$ by $10$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{- 10 \sqrt{2}}{2^{3}} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.23

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

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{- 10 \sqrt{2}}{8} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.24

Cancel the common factor of $- 10$ and $8$ .

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Step 3.1.24.1

Factor $2$ out of $- 10 \sqrt{2}$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{2 (- 5 \sqrt{2})}{8} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.24.2

Cancel the common factors.

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Step 3.1.24.2.1

Factor $2$ out of $8$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{2 (- 5 \sqrt{2})}{2 (4)} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.24.2.2

Cancel the common factor.

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{2 (- 5 \sqrt{2})}{2 \cdot 4} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.24.2.3

Rewrite the expression.

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} + \frac{- 5 \sqrt{2}}{4} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.25

Move the negative in front of the fraction.

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{(5) \sqrt{2}}{4} + 10 {(\frac{\sqrt{2}}{2})}^{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.26

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

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 10 \frac{{\sqrt{2}}^{2}}{2^{2}} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.27

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

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Step 3.1.27.1

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

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 10 \frac{{(2^{\frac{1}{2}})}^{2}}{2^{2}} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.27.2

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

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 10 \frac{2^{\frac{1}{2} \cdot 2}}{2^{2}} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.27.3

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

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 10 \frac{2^{\frac{2}{2}}}{2^{2}} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.27.4

Cancel the common factor of $2$ .

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Step 3.1.27.4.1

Cancel the common factor.

Step 3.1.27.4.2

Rewrite the expression.

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 10 \frac{2^{1}}{2^{2}} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.27.5

Evaluate the exponent.

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 10 \frac{2}{2^{2}} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.28

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

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 10 (\frac{2}{4}) {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.29

Cancel the common factor of $2$ .

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Step 3.1.29.1

Factor $2$ out of $10$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 2 (5) \frac{2}{4} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.29.2

Factor $2$ out of $4$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 2 \cdot 5 \frac{2}{2 \cdot 2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.29.3

Cancel the common factor.

Step 3.1.29.4

Rewrite the expression.

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 5 (\frac{2}{2}) {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.30

Combine $5$ and $\frac{2}{2}$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + \frac{5 \cdot 2}{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.31

Multiply $5$ by $2$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + \frac{10}{2} {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.32

Divide $10$ by $2$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 5 {(\frac{i \sqrt{2}}{2})}^{3} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.33

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

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Step 3.1.33.1

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

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 5 \frac{{(i \sqrt{2})}^{3}}{2^{3}} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.33.2

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

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 5 \frac{i^{3} {\sqrt{2}}^{3}}{2^{3}} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.34

Simplify the numerator.

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Step 3.1.34.1

Factor out $i^{2}$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 5 \frac{i^{2} \cdot i {\sqrt{2}}^{3}}{2^{3}} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.34.2

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

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 5 \frac{- 1 \cdot i {\sqrt{2}}^{3}}{2^{3}} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.34.3

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

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 5 \frac{- i {\sqrt{2}}^{3}}{2^{3}} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.34.4

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

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 5 \frac{- i \sqrt{2^{3}}}{2^{3}} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.34.5

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

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 5 \frac{- i \sqrt{8}}{2^{3}} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.34.6

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

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Step 3.1.34.6.1

Factor $4$ out of $8$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 5 \frac{- i \sqrt{4 (2)}}{2^{3}} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.34.6.2

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

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 5 \frac{- i \sqrt{2^{2} \cdot 2}}{2^{3}} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.34.7

Pull terms out from under the radical.

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 5 \frac{- i \cdot 2 \sqrt{2}}{2^{3}} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.34.8

Multiply $2$ by $- 1$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 5 \frac{- 2 i \sqrt{2}}{2^{3}} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.35

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

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 5 \frac{- 2 i \sqrt{2}}{8} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.36

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

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Step 3.1.36.1

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

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 5 \frac{2 (- i \sqrt{2})}{8} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.36.2

Cancel the common factors.

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Step 3.1.36.2.1

Factor $2$ out of $8$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 5 \frac{2 (- i \sqrt{2})}{2 (4)} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.36.2.2

Cancel the common factor.

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 5 \frac{2 (- i \sqrt{2})}{2 \cdot 4} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.36.2.3

Rewrite the expression.

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 5 \frac{- i \sqrt{2}}{4} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.37

Move the negative in front of the fraction.

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + 5 (- \frac{i \sqrt{2}}{4}) + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.38

Multiply $5 (- \frac{i \sqrt{2}}{4})$ .

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Step 3.1.38.1

Multiply $- 1$ by $5$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - 5 \frac{i \sqrt{2}}{4} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.38.2

Combine $- 5$ and $\frac{i \sqrt{2}}{4}$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} + \frac{- 5 (i \sqrt{2})}{4} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.39

Move the negative in front of the fraction.

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{(5) i \sqrt{2}}{4} + 5 \frac{\sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.40

Combine $5$ and $\frac{\sqrt{2}}{2}$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{2} {(\frac{i \sqrt{2}}{2})}^{4} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.41

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

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Step 3.1.41.1

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

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{2} \cdot \frac{{(i \sqrt{2})}^{4}}{2^{4}} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.41.2

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

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{2} \cdot \frac{i^{4} {\sqrt{2}}^{4}}{2^{4}} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.42

Combine.

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2} (i^{4} {\sqrt{2}}^{4})}{2 \cdot 2^{4}} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.43

Multiply $\sqrt{2}$ by ${\sqrt{2}}^{4}$ by adding the exponents.

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Step 3.1.43.1

Move ${\sqrt{2}}^{4}$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 ({\sqrt{2}}^{4} \sqrt{2}) i^{4}}{2 \cdot 2^{4}} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.43.2

Multiply ${\sqrt{2}}^{4}$ by $\sqrt{2}$ .

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Step 3.1.43.2.1

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

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 ({\sqrt{2}}^{4} {\sqrt{2}}^{1}) i^{4}}{2 \cdot 2^{4}} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.43.2.2

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

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 {\sqrt{2}}^{4 + 1} i^{4}}{2 \cdot 2^{4}} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.43.3

Add $4$ and $1$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 {\sqrt{2}}^{5} i^{4}}{2 \cdot 2^{4}} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.44

Multiply $2$ by $2^{4}$ by adding the exponents.

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Step 3.1.44.1

Multiply $2$ by $2^{4}$ .

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Step 3.1.44.1.1

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

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 {\sqrt{2}}^{5} i^{4}}{2^{1} \cdot 2^{4}} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.44.1.2

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

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 {\sqrt{2}}^{5} i^{4}}{2^{1 + 4}} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.44.2

Add $1$ and $4$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 {\sqrt{2}}^{5} i^{4}}{2^{5}} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.45

Simplify the numerator.

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Step 3.1.45.1

Rewrite ${\sqrt{2}}^{5}$ as $\sqrt{2^{5}}$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2^{5}} i^{4}}{2^{5}} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.45.2

Raise $2$ to the power of $5$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{32} i^{4}}{2^{5}} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.45.3

Rewrite $32$ as $4^{2} \cdot 2$ .

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Step 3.1.45.3.1

Factor $16$ out of $32$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{16 (2)} i^{4}}{2^{5}} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.45.3.2

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

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{4^{2} \cdot 2} i^{4}}{2^{5}} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.45.4

Pull terms out from under the radical.

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 (4 \sqrt{2}) i^{4}}{2^{5}} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.45.5

Rewrite $i^{4}$ as $1$ .

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Step 3.1.45.5.1

Rewrite $i^{4}$ as ${(i^{2})}^{2}$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 (4 \sqrt{2}) {(i^{2})}^{2}}{2^{5}} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.45.5.2

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

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 (4 \sqrt{2}) {(- 1)}^{2}}{2^{5}} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.45.5.3

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

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 (4 \sqrt{2}) \cdot 1}{2^{5}} + {(\frac{i \sqrt{2}}{2})}^{5}$

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \cdot 4 \sqrt{2} \cdot 1}{2^{5}} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.45.6

Combine exponents.

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Step 3.1.45.6.1

Multiply $5$ by $4$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{20 \sqrt{2} \cdot 1}{2^{5}} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.45.6.2

Multiply $20$ by $1$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{20 \sqrt{2}}{2^{5}} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.46

Raise $2$ to the power of $5$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{20 \sqrt{2}}{32} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.47

Cancel the common factor of $20$ and $32$ .

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Step 3.1.47.1

Factor $4$ out of $20 \sqrt{2}$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{4 (5 \sqrt{2})}{32} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.47.2

Cancel the common factors.

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Step 3.1.47.2.1

Factor $4$ out of $32$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{4 (5 \sqrt{2})}{4 (8)} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.47.2.2

Cancel the common factor.

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{4 (5 \sqrt{2})}{4 \cdot 8} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.47.2.3

Rewrite the expression.

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{8} + {(\frac{i \sqrt{2}}{2})}^{5}$

Step 3.1.48

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

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Step 3.1.48.1

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

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{8} + \frac{{(i \sqrt{2})}^{5}}{2^{5}}$

Step 3.1.48.2

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

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{8} + \frac{i^{5} {\sqrt{2}}^{5}}{2^{5}}$

Step 3.1.49

Simplify the numerator.

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Step 3.1.49.1

Factor out $i^{4}$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{8} + \frac{i^{4} i {\sqrt{2}}^{5}}{2^{5}}$

Step 3.1.49.2

Rewrite $i^{4}$ as $1$ .

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Step 3.1.49.2.1

Rewrite $i^{4}$ as ${(i^{2})}^{2}$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{8} + \frac{{(i^{2})}^{2} i {\sqrt{2}}^{5}}{2^{5}}$

Step 3.1.49.2.2

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

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{8} + \frac{{(- 1)}^{2} i {\sqrt{2}}^{5}}{2^{5}}$

Step 3.1.49.2.3

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

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{8} + \frac{1 i {\sqrt{2}}^{5}}{2^{5}}$

Step 3.1.49.3

Multiply $i$ by $1$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{8} + \frac{i {\sqrt{2}}^{5}}{2^{5}}$

Step 3.1.49.4

Rewrite ${\sqrt{2}}^{5}$ as $\sqrt{2^{5}}$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{8} + \frac{i \sqrt{2^{5}}}{2^{5}}$

Step 3.1.49.5

Raise $2$ to the power of $5$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{8} + \frac{i \sqrt{32}}{2^{5}}$

Step 3.1.49.6

Rewrite $32$ as $4^{2} \cdot 2$ .

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Step 3.1.49.6.1

Factor $16$ out of $32$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{8} + \frac{i \sqrt{16 (2)}}{2^{5}}$

Step 3.1.49.6.2

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

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{8} + \frac{i \sqrt{4^{2} \cdot 2}}{2^{5}}$

Step 3.1.49.7

Pull terms out from under the radical.

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{8} + \frac{i \cdot 4 \sqrt{2}}{2^{5}}$

Step 3.1.50

Raise $2$ to the power of $5$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{8} + \frac{i \cdot 4 \sqrt{2}}{32}$

Step 3.1.51

Move $4$ to the left of $i$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{8} + \frac{4 i \sqrt{2}}{32}$

Step 3.1.52

Cancel the common factor of $4$ and $32$ .

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Step 3.1.52.1

Factor $4$ out of $4 i \sqrt{2}$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{8} + \frac{4 (i \sqrt{2})}{32}$

Step 3.1.52.2

Cancel the common factors.

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Step 3.1.52.2.1

Factor $4$ out of $32$ .

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{8} + \frac{4 (i \sqrt{2})}{4 \cdot 8}$

Step 3.1.52.2.2

Cancel the common factor.

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{8} + \frac{4 (i \sqrt{2})}{4 \cdot 8}$

Step 3.1.52.2.3

Rewrite the expression.

$\frac{\sqrt{2}}{8} + \frac{5 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4} + \frac{5 \sqrt{2}}{8} + \frac{i \sqrt{2}}{8}$

Step 3.2

Simplify terms.

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Step 3.2.1

Combine the numerators over the common denominator.

$\frac{\sqrt{2} + 5 i \sqrt{2} + 5 \sqrt{2} + i \sqrt{2}}{8} + \frac{- 5 \sqrt{2}}{4} + \frac{- 5 i \sqrt{2}}{4}$

Step 3.2.2

Add $\sqrt{2}$ and $5 \sqrt{2}$ .

$\frac{5 i \sqrt{2} + 6 \sqrt{2} + i \sqrt{2}}{8} + \frac{- 5 \sqrt{2}}{4} + \frac{- 5 i \sqrt{2}}{4}$

Step 3.2.3

Add $5 i \sqrt{2}$ and $i \sqrt{2}$ .

$\frac{6 i \sqrt{2} + 6 \sqrt{2}}{8} + \frac{- 5 \sqrt{2}}{4} + \frac{- 5 i \sqrt{2}}{4}$

Step 3.2.4

Reorder $6 i \sqrt{2}$ and $6 \sqrt{2}$ .

$\frac{6 \sqrt{2} + 6 i \sqrt{2}}{8} + \frac{- 5 \sqrt{2}}{4} + \frac{- 5 i \sqrt{2}}{4}$

Step 3.3

Simplify each term.

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Step 3.3.1

Move the negative in front of the fraction.

$\frac{6 \sqrt{2} + 6 i \sqrt{2}}{8} - \frac{(5) \sqrt{2}}{4} + \frac{- 5 i \sqrt{2}}{4}$

Step 3.3.2

Move the negative in front of the fraction.

$\frac{6 \sqrt{2} + 6 i \sqrt{2}}{8} - \frac{5 \sqrt{2}}{4} - \frac{5 i \sqrt{2}}{4}$

Step 3.4

Reorder $\frac{6 \sqrt{2} + 6 i \sqrt{2}}{8}$ and $- \frac{5 \sqrt{2}}{4}$ .

$- \frac{5 \sqrt{2}}{4} + \frac{6 \sqrt{2} + 6 i \sqrt{2}}{8} - \frac{5 i \sqrt{2}}{4}$

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 $a = - \frac{5 \sqrt{2}}{4}$ and $b = \frac{1}{8}$ .

$| z | = \sqrt{{(\frac{1}{8})}^{2} + {(- \frac{5 \sqrt{2}}{4})}^{2}}$

Step 7

Find $| z |$ .

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Step 7.1

Apply the product rule to $\frac{1}{8}$ .

$| z | = \sqrt{\frac{1^{2}}{8^{2}} + {(- \frac{5 \sqrt{2}}{4})}^{2}}$

Step 7.2

One to any power is one.

$| z | = \sqrt{\frac{1}{8^{2}} + {(- \frac{5 \sqrt{2}}{4})}^{2}}$

Step 7.3

Raise $8$ to the power of $2$ .

$| z | = \sqrt{\frac{1}{64} + {(- \frac{5 \sqrt{2}}{4})}^{2}}$

Step 7.4

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

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Step 7.4.1

Apply the product rule to $- \frac{5 \sqrt{2}}{4}$ .

$| z | = \sqrt{\frac{1}{64} + {(- 1)}^{2} {(\frac{5 \sqrt{2}}{4})}^{2}}$

Step 7.4.2

Apply the product rule to $\frac{5 \sqrt{2}}{4}$ .

$| z | = \sqrt{\frac{1}{64} + {(- 1)}^{2} (\frac{{(5 \sqrt{2})}^{2}}{4^{2}})}$

Step 7.4.3

Apply the product rule to $5 \sqrt{2}$ .

$| z | = \sqrt{\frac{1}{64} + {(- 1)}^{2} (\frac{5^{2} {\sqrt{2}}^{2}}{4^{2}})}$

Step 7.5

Simplify the expression.

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Step 7.5.1

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

$| z | = \sqrt{\frac{1}{64} + 1 (\frac{5^{2} {\sqrt{2}}^{2}}{4^{2}})}$

Step 7.5.2

Multiply $\frac{5^{2} {\sqrt{2}}^{2}}{4^{2}}$ by $1$ .

$| z | = \sqrt{\frac{1}{64} + \frac{5^{2} {\sqrt{2}}^{2}}{4^{2}}}$

Step 7.6

Simplify the numerator.

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Step 7.6.1

Raise $5$ to the power of $2$ .

$| z | = \sqrt{\frac{1}{64} + \frac{25 {\sqrt{2}}^{2}}{4^{2}}}$

Step 7.6.2

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

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Step 7.6.2.1

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

$| z | = \sqrt{\frac{1}{64} + \frac{25 {(2^{\frac{1}{2}})}^{2}}{4^{2}}}$

Step 7.6.2.2

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

$| z | = \sqrt{\frac{1}{64} + \frac{25 \cdot 2^{\frac{1}{2} \cdot 2}}{4^{2}}}$

Step 7.6.2.3

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

$| z | = \sqrt{\frac{1}{64} + \frac{25 \cdot 2^{\frac{2}{2}}}{4^{2}}}$

Step 7.6.2.4

Cancel the common factor of $2$ .

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Step 7.6.2.4.1

Cancel the common factor.

$| z | = \sqrt{\frac{1}{64} + \frac{25 \cdot 2^{\frac{2}{2}}}{4^{2}}}$

Step 7.6.2.4.2

Rewrite the expression.

$| z | = \sqrt{\frac{1}{64} + \frac{25 \cdot 2}{4^{2}}}$

Step 7.6.2.5

Evaluate the exponent.

$| z | = \sqrt{\frac{1}{64} + \frac{25 \cdot 2}{4^{2}}}$

Step 7.7

Reduce the expression by cancelling the common factors.

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Step 7.7.1

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

$| z | = \sqrt{\frac{1}{64} + \frac{25 \cdot 2}{16}}$

Step 7.7.2

Multiply $25$ by $2$ .

$| z | = \sqrt{\frac{1}{64} + \frac{50}{16}}$

Step 7.7.3

Cancel the common factor of $50$ and $16$ .

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Step 7.7.3.1

Factor $2$ out of $50$ .

$| z | = \sqrt{\frac{1}{64} + \frac{2 (25)}{16}}$

Step 7.7.3.2

Cancel the common factors.

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Step 7.7.3.2.1

Factor $2$ out of $16$ .

$| z | = \sqrt{\frac{1}{64} + \frac{2 \cdot 25}{2 \cdot 8}}$

Step 7.7.3.2.2

Cancel the common factor.

$| z | = \sqrt{\frac{1}{64} + \frac{2 \cdot 25}{2 \cdot 8}}$

Step 7.7.3.2.3

Rewrite the expression.

$| z | = \sqrt{\frac{1}{64} + \frac{25}{8}}$

Step 7.8

To write $\frac{25}{8}$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$| z | = \sqrt{\frac{1}{64} + \frac{25}{8} \cdot \frac{8}{8}}$

Step 7.9

Write each expression with a common denominator of $64$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.9.1

Multiply $\frac{25}{8}$ by $\frac{8}{8}$ .

$| z | = \sqrt{\frac{1}{64} + \frac{25 \cdot 8}{8 \cdot 8}}$

Step 7.9.2

Multiply $8$ by $8$ .

$| z | = \sqrt{\frac{1}{64} + \frac{25 \cdot 8}{64}}$

Step 7.10

Combine the numerators over the common denominator.

$| z | = \sqrt{\frac{1 + 25 \cdot 8}{64}}$

Step 7.11

Simplify the numerator.

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Step 7.11.1

Multiply $25$ by $8$ .

$| z | = \sqrt{\frac{1 + 200}{64}}$

Step 7.11.2

Add $1$ and $200$ .

$| z | = \sqrt{\frac{201}{64}}$

Step 7.12

Rewrite $\sqrt{\frac{201}{64}}$ as $\frac{\sqrt{201}}{\sqrt{64}}$ .

$| z | = \frac{\sqrt{201}}{\sqrt{64}}$

Step 7.13

Simplify the denominator.

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Step 7.13.1

Rewrite $64$ as $8^{2}$ .

$| z | = \frac{\sqrt{201}}{\sqrt{8^{2}}}$

Step 7.13.2

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

$| z | = \frac{\sqrt{201}}{8}$

Step 8

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

$θ = \arctan (\frac{\frac{1}{8}}{- \frac{5 \sqrt{2}}{4}})$

Step 9

Since inverse tangent of $\frac{\frac{1}{8}}{- \frac{5 \sqrt{2}}{4}}$ produces an angle in the second quadrant, the value of the angle is $3.07099947$ .

$θ = 3.07099947$

Step 10

Substitute the values of $θ = 3.07099947$ and $| z | = \frac{\sqrt{201}}{8}$ .

$\frac{\sqrt{201}}{8 (\cos (3.07099947) + i \sin (3.07099947))}$