Precalculus Examples

${(\sqrt{2} - i)}^{6}$

Step 1

Use the Binomial Theorem.

${\sqrt{2}}^{6} + 6 {\sqrt{2}}^{5} (- i) + 15 {\sqrt{2}}^{4} {(- i)}^{2} + 20 {\sqrt{2}}^{3} {(- i)}^{3} + 15 {\sqrt{2}}^{2} {(- i)}^{4} + 6 \sqrt{2} {(- i)}^{5} + {(- i)}^{6}$

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite ${\sqrt{2}}^{6}$ as $2^{3}$ .

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Step 2.1.1.1

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

${(2^{\frac{1}{2}})}^{6} + 6 {\sqrt{2}}^{5} (- i) + 15 {\sqrt{2}}^{4} {(- i)}^{2} + 20 {\sqrt{2}}^{3} {(- i)}^{3} + 15 {\sqrt{2}}^{2} {(- i)}^{4} + 6 \sqrt{2} {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.1.2

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

$2^{\frac{1}{2} \cdot 6} + 6 {\sqrt{2}}^{5} (- i) + 15 {\sqrt{2}}^{4} {(- i)}^{2} + 20 {\sqrt{2}}^{3} {(- i)}^{3} + 15 {\sqrt{2}}^{2} {(- i)}^{4} + 6 \sqrt{2} {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.1.3

Combine $\frac{1}{2}$ and $6$ .

$2^{\frac{6}{2}} + 6 {\sqrt{2}}^{5} (- i) + 15 {\sqrt{2}}^{4} {(- i)}^{2} + 20 {\sqrt{2}}^{3} {(- i)}^{3} + 15 {\sqrt{2}}^{2} {(- i)}^{4} + 6 \sqrt{2} {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.1.4

Cancel the common factor of $6$ and $2$ .

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Step 2.1.1.4.1

Factor $2$ out of $6$ .

$2^{\frac{2 \cdot 3}{2}} + 6 {\sqrt{2}}^{5} (- i) + 15 {\sqrt{2}}^{4} {(- i)}^{2} + 20 {\sqrt{2}}^{3} {(- i)}^{3} + 15 {\sqrt{2}}^{2} {(- i)}^{4} + 6 \sqrt{2} {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.1.4.2

Cancel the common factors.

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Step 2.1.1.4.2.1

Factor $2$ out of $2$ .

$2^{\frac{2 \cdot 3}{2 (1)}} + 6 {\sqrt{2}}^{5} (- i) + 15 {\sqrt{2}}^{4} {(- i)}^{2} + 20 {\sqrt{2}}^{3} {(- i)}^{3} + 15 {\sqrt{2}}^{2} {(- i)}^{4} + 6 \sqrt{2} {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.1.4.2.2

Cancel the common factor.

$2^{\frac{2 \cdot 3}{2 \cdot 1}} + 6 {\sqrt{2}}^{5} (- i) + 15 {\sqrt{2}}^{4} {(- i)}^{2} + 20 {\sqrt{2}}^{3} {(- i)}^{3} + 15 {\sqrt{2}}^{2} {(- i)}^{4} + 6 \sqrt{2} {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.1.4.2.3

Rewrite the expression.

$2^{\frac{3}{1}} + 6 {\sqrt{2}}^{5} (- i) + 15 {\sqrt{2}}^{4} {(- i)}^{2} + 20 {\sqrt{2}}^{3} {(- i)}^{3} + 15 {\sqrt{2}}^{2} {(- i)}^{4} + 6 \sqrt{2} {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.1.4.2.4

Divide $3$ by $1$ .

$2^{3} + 6 {\sqrt{2}}^{5} (- i) + 15 {\sqrt{2}}^{4} {(- i)}^{2} + 20 {\sqrt{2}}^{3} {(- i)}^{3} + 15 {\sqrt{2}}^{2} {(- i)}^{4} + 6 \sqrt{2} {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.2

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

$8 + 6 {\sqrt{2}}^{5} (- i) + 15 {\sqrt{2}}^{4} {(- i)}^{2} + 20 {\sqrt{2}}^{3} {(- i)}^{3} + 15 {\sqrt{2}}^{2} {(- i)}^{4} + 6 \sqrt{2} {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.3

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

$8 + 6 \sqrt{2^{5}} (- i) + 15 {\sqrt{2}}^{4} {(- i)}^{2} + 20 {\sqrt{2}}^{3} {(- i)}^{3} + 15 {\sqrt{2}}^{2} {(- i)}^{4} + 6 \sqrt{2} {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.4

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

$8 + 6 \sqrt{32} (- i) + 15 {\sqrt{2}}^{4} {(- i)}^{2} + 20 {\sqrt{2}}^{3} {(- i)}^{3} + 15 {\sqrt{2}}^{2} {(- i)}^{4} + 6 \sqrt{2} {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.5

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

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Step 2.1.5.1

Factor $16$ out of $32$ .

$8 + 6 \sqrt{16 (2)} (- i) + 15 {\sqrt{2}}^{4} {(- i)}^{2} + 20 {\sqrt{2}}^{3} {(- i)}^{3} + 15 {\sqrt{2}}^{2} {(- i)}^{4} + 6 \sqrt{2} {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.5.2

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

$8 + 6 \sqrt{4^{2} \cdot 2} (- i) + 15 {\sqrt{2}}^{4} {(- i)}^{2} + 20 {\sqrt{2}}^{3} {(- i)}^{3} + 15 {\sqrt{2}}^{2} {(- i)}^{4} + 6 \sqrt{2} {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.6

Pull terms out from under the radical.

$8 + 6 (4 \sqrt{2}) (- i) + 15 {\sqrt{2}}^{4} {(- i)}^{2} + 20 {\sqrt{2}}^{3} {(- i)}^{3} + 15 {\sqrt{2}}^{2} {(- i)}^{4} + 6 \sqrt{2} {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.7

Multiply $4$ by $6$ .

$8 + 24 \sqrt{2} (- i) + 15 {\sqrt{2}}^{4} {(- i)}^{2} + 20 {\sqrt{2}}^{3} {(- i)}^{3} + 15 {\sqrt{2}}^{2} {(- i)}^{4} + 6 \sqrt{2} {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.8

Multiply $- 1$ by $24$ .

$8 - 24 \sqrt{2} i + 15 {\sqrt{2}}^{4} {(- i)}^{2} + 20 {\sqrt{2}}^{3} {(- i)}^{3} + 15 {\sqrt{2}}^{2} {(- i)}^{4} + 6 \sqrt{2} {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.9

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

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Step 2.1.9.1

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

$8 - 24 \sqrt{2} i + 15 {(2^{\frac{1}{2}})}^{4} {(- i)}^{2} + 20 {\sqrt{2}}^{3} {(- i)}^{3} + 15 {\sqrt{2}}^{2} {(- i)}^{4} + 6 \sqrt{2} {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.9.2

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

$8 - 24 \sqrt{2} i + 15 \cdot 2^{\frac{1}{2} \cdot 4} {(- i)}^{2} + 20 {\sqrt{2}}^{3} {(- i)}^{3} + 15 {\sqrt{2}}^{2} {(- i)}^{4} + 6 \sqrt{2} {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.9.3

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

$8 - 24 \sqrt{2} i + 15 \cdot 2^{\frac{4}{2}} {(- i)}^{2} + 20 {\sqrt{2}}^{3} {(- i)}^{3} + 15 {\sqrt{2}}^{2} {(- i)}^{4} + 6 \sqrt{2} {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.9.4

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

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Step 2.1.9.4.1

Factor $2$ out of $4$ .

$8 - 24 \sqrt{2} i + 15 \cdot 2^{\frac{2 \cdot 2}{2}} {(- i)}^{2} + 20 {\sqrt{2}}^{3} {(- i)}^{3} + 15 {\sqrt{2}}^{2} {(- i)}^{4} + 6 \sqrt{2} {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.9.4.2

Cancel the common factors.

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Step 2.1.9.4.2.1

Factor $2$ out of $2$ .

$8 - 24 \sqrt{2} i + 15 \cdot 2^{\frac{2 \cdot 2}{2 (1)}} {(- i)}^{2} + 20 {\sqrt{2}}^{3} {(- i)}^{3} + 15 {\sqrt{2}}^{2} {(- i)}^{4} + 6 \sqrt{2} {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.9.4.2.2

Cancel the common factor.

$8 - 24 \sqrt{2} i + 15 \cdot 2^{\frac{2 \cdot 2}{2 \cdot 1}} {(- i)}^{2} + 20 {\sqrt{2}}^{3} {(- i)}^{3} + 15 {\sqrt{2}}^{2} {(- i)}^{4} + 6 \sqrt{2} {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.9.4.2.3

Rewrite the expression.

$8 - 24 \sqrt{2} i + 15 \cdot 2^{\frac{2}{1}} {(- i)}^{2} + 20 {\sqrt{2}}^{3} {(- i)}^{3} + 15 {\sqrt{2}}^{2} {(- i)}^{4} + 6 \sqrt{2} {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.9.4.2.4

Divide $2$ by $1$ .

$8 - 24 \sqrt{2} i + 15 \cdot 2^{2} {(- i)}^{2} + 20 {\sqrt{2}}^{3} {(- i)}^{3} + 15 {\sqrt{2}}^{2} {(- i)}^{4} + 6 \sqrt{2} {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.10

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

$8 - 24 \sqrt{2} i + 15 \cdot 4 {(- i)}^{2} + 20 {\sqrt{2}}^{3} {(- i)}^{3} + 15 {\sqrt{2}}^{2} {(- i)}^{4} + 6 \sqrt{2} {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.11

Multiply $15$ by $4$ .

$8 - 24 \sqrt{2} i + 60 {(- i)}^{2} + 20 {\sqrt{2}}^{3} {(- i)}^{3} + 15 {\sqrt{2}}^{2} {(- i)}^{4} + 6 \sqrt{2} {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.12

Apply the product rule to $- i$ .

$8 - 24 \sqrt{2} i + 60 ({(- 1)}^{2} i^{2}) + 20 {\sqrt{2}}^{3} {(- i)}^{3} + 15 {\sqrt{2}}^{2} {(- i)}^{4} + 6 \sqrt{2} {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.13

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

$8 - 24 \sqrt{2} i + 60 (1 i^{2}) + 20 {\sqrt{2}}^{3} {(- i)}^{3} + 15 {\sqrt{2}}^{2} {(- i)}^{4} + 6 \sqrt{2} {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.14

Multiply $i^{2}$ by $1$ .

$8 - 24 \sqrt{2} i + 60 i^{2} + 20 {\sqrt{2}}^{3} {(- i)}^{3} + 15 {\sqrt{2}}^{2} {(- i)}^{4} + 6 \sqrt{2} {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.15

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

$8 - 24 \sqrt{2} i + 60 \cdot - 1 + 20 {\sqrt{2}}^{3} {(- i)}^{3} + 15 {\sqrt{2}}^{2} {(- i)}^{4} + 6 \sqrt{2} {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.16

Multiply $60$ by $- 1$ .

$8 - 24 \sqrt{2} i - 60 + 20 {\sqrt{2}}^{3} {(- i)}^{3} + 15 {\sqrt{2}}^{2} {(- i)}^{4} + 6 \sqrt{2} {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.17

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

$8 - 24 \sqrt{2} i - 60 + 20 \sqrt{2^{3}} {(- i)}^{3} + 15 {\sqrt{2}}^{2} {(- i)}^{4} + 6 \sqrt{2} {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.18

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

$8 - 24 \sqrt{2} i - 60 + 20 \sqrt{8} {(- i)}^{3} + 15 {\sqrt{2}}^{2} {(- i)}^{4} + 6 \sqrt{2} {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.19

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

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Step 2.1.19.1

Factor $4$ out of $8$ .

$8 - 24 \sqrt{2} i - 60 + 20 \sqrt{4 (2)} {(- i)}^{3} + 15 {\sqrt{2}}^{2} {(- i)}^{4} + 6 \sqrt{2} {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.19.2

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

$8 - 24 \sqrt{2} i - 60 + 20 \sqrt{2^{2} \cdot 2} {(- i)}^{3} + 15 {\sqrt{2}}^{2} {(- i)}^{4} + 6 \sqrt{2} {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.20

Pull terms out from under the radical.

$8 - 24 \sqrt{2} i - 60 + 20 (2 \sqrt{2}) {(- i)}^{3} + 15 {\sqrt{2}}^{2} {(- i)}^{4} + 6 \sqrt{2} {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.21

Multiply $2$ by $20$ .

$8 - 24 \sqrt{2} i - 60 + 40 \sqrt{2} {(- i)}^{3} + 15 {\sqrt{2}}^{2} {(- i)}^{4} + 6 \sqrt{2} {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.22

Apply the product rule to $- i$ .

$8 - 24 \sqrt{2} i - 60 + 40 \sqrt{2} ({(- 1)}^{3} i^{3}) + 15 {\sqrt{2}}^{2} {(- i)}^{4} + 6 \sqrt{2} {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.23

Raise $- 1$ to the power of $3$ .

$8 - 24 \sqrt{2} i - 60 + 40 \sqrt{2} (- i^{3}) + 15 {\sqrt{2}}^{2} {(- i)}^{4} + 6 \sqrt{2} {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.24

Factor out $i^{2}$ .

$8 - 24 \sqrt{2} i - 60 + 40 \sqrt{2} (- (i^{2} \cdot i)) + 15 {\sqrt{2}}^{2} {(- i)}^{4} + 6 \sqrt{2} {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.25

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

$8 - 24 \sqrt{2} i - 60 + 40 \sqrt{2} (- (- 1 \cdot i)) + 15 {\sqrt{2}}^{2} {(- i)}^{4} + 6 \sqrt{2} {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.26

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

$8 - 24 \sqrt{2} i - 60 + 40 \sqrt{2} (- - i) + 15 {\sqrt{2}}^{2} {(- i)}^{4} + 6 \sqrt{2} {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.27

Multiply $- 1$ by $- 1$ .

$8 - 24 \sqrt{2} i - 60 + 40 \sqrt{2} (1 i) + 15 {\sqrt{2}}^{2} {(- i)}^{4} + 6 \sqrt{2} {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.28

Multiply $i$ by $1$ .

$8 - 24 \sqrt{2} i - 60 + 40 \sqrt{2} i + 15 {\sqrt{2}}^{2} {(- i)}^{4} + 6 \sqrt{2} {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.29

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

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Step 2.1.29.1

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

$8 - 24 \sqrt{2} i - 60 + 40 \sqrt{2} i + 15 {(2^{\frac{1}{2}})}^{2} {(- i)}^{4} + 6 \sqrt{2} {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.29.2

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

$8 - 24 \sqrt{2} i - 60 + 40 \sqrt{2} i + 15 \cdot 2^{\frac{1}{2} \cdot 2} {(- i)}^{4} + 6 \sqrt{2} {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.29.3

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

$8 - 24 \sqrt{2} i - 60 + 40 \sqrt{2} i + 15 \cdot 2^{\frac{2}{2}} {(- i)}^{4} + 6 \sqrt{2} {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.29.4

Cancel the common factor of $2$ .

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Step 2.1.29.4.1

Cancel the common factor.

$8 - 24 \sqrt{2} i - 60 + 40 \sqrt{2} i + 15 \cdot 2^{\frac{2}{2}} {(- i)}^{4} + 6 \sqrt{2} {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.29.4.2

Rewrite the expression.

$8 - 24 \sqrt{2} i - 60 + 40 \sqrt{2} i + 15 \cdot 2^{1} {(- i)}^{4} + 6 \sqrt{2} {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.29.5

Evaluate the exponent.

$8 - 24 \sqrt{2} i - 60 + 40 \sqrt{2} i + 15 \cdot 2 {(- i)}^{4} + 6 \sqrt{2} {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.30

Multiply $15$ by $2$ .

$8 - 24 \sqrt{2} i - 60 + 40 \sqrt{2} i + 30 {(- i)}^{4} + 6 \sqrt{2} {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.31

Apply the product rule to $- i$ .

$8 - 24 \sqrt{2} i - 60 + 40 \sqrt{2} i + 30 ({(- 1)}^{4} i^{4}) + 6 \sqrt{2} {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.32

Raise $- 1$ to the power of $4$ .

$8 - 24 \sqrt{2} i - 60 + 40 \sqrt{2} i + 30 (1 i^{4}) + 6 \sqrt{2} {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.33

Multiply $i^{4}$ by $1$ .

$8 - 24 \sqrt{2} i - 60 + 40 \sqrt{2} i + 30 i^{4} + 6 \sqrt{2} {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.34

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

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Step 2.1.34.1

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

$8 - 24 \sqrt{2} i - 60 + 40 \sqrt{2} i + 30 {(i^{2})}^{2} + 6 \sqrt{2} {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.34.2

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

$8 - 24 \sqrt{2} i - 60 + 40 \sqrt{2} i + 30 {(- 1)}^{2} + 6 \sqrt{2} {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.34.3

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

$8 - 24 \sqrt{2} i - 60 + 40 \sqrt{2} i + 30 \cdot 1 + 6 \sqrt{2} {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.35

Multiply $30$ by $1$ .

$8 - 24 \sqrt{2} i - 60 + 40 \sqrt{2} i + 30 + 6 \sqrt{2} {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.36

Apply the product rule to $- i$ .

$8 - 24 \sqrt{2} i - 60 + 40 \sqrt{2} i + 30 + 6 \sqrt{2} ({(- 1)}^{5} i^{5}) + {(- i)}^{6}$

Step 2.1.37

Raise $- 1$ to the power of $5$ .

$8 - 24 \sqrt{2} i - 60 + 40 \sqrt{2} i + 30 + 6 \sqrt{2} (- i^{5}) + {(- i)}^{6}$

Step 2.1.38

Factor out $i^{4}$ .

$8 - 24 \sqrt{2} i - 60 + 40 \sqrt{2} i + 30 + 6 \sqrt{2} (- (i^{4} i)) + {(- i)}^{6}$

Step 2.1.39

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

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Step 2.1.39.1

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

$8 - 24 \sqrt{2} i - 60 + 40 \sqrt{2} i + 30 + 6 \sqrt{2} (- ({(i^{2})}^{2} i)) + {(- i)}^{6}$

Step 2.1.39.2

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

$8 - 24 \sqrt{2} i - 60 + 40 \sqrt{2} i + 30 + 6 \sqrt{2} (- ({(- 1)}^{2} i)) + {(- i)}^{6}$

Step 2.1.39.3

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

$8 - 24 \sqrt{2} i - 60 + 40 \sqrt{2} i + 30 + 6 \sqrt{2} (- (1 i)) + {(- i)}^{6}$

Step 2.1.40

Multiply $i$ by $1$ .

$8 - 24 \sqrt{2} i - 60 + 40 \sqrt{2} i + 30 + 6 \sqrt{2} (- i) + {(- i)}^{6}$

Step 2.1.41

Multiply $- 1$ by $6$ .

$8 - 24 \sqrt{2} i - 60 + 40 \sqrt{2} i + 30 - 6 \sqrt{2} i + {(- i)}^{6}$

Step 2.1.42

Apply the product rule to $- i$ .

$8 - 24 \sqrt{2} i - 60 + 40 \sqrt{2} i + 30 - 6 \sqrt{2} i + {(- 1)}^{6} i^{6}$

Step 2.1.43

Raise $- 1$ to the power of $6$ .

$8 - 24 \sqrt{2} i - 60 + 40 \sqrt{2} i + 30 - 6 \sqrt{2} i + 1 i^{6}$

Step 2.1.44

Multiply $i^{6}$ by $1$ .

$8 - 24 \sqrt{2} i - 60 + 40 \sqrt{2} i + 30 - 6 \sqrt{2} i + i^{6}$

Step 2.1.45

Factor out $i^{4}$ .

$8 - 24 \sqrt{2} i - 60 + 40 \sqrt{2} i + 30 - 6 \sqrt{2} i + i^{4} i^{2}$

Step 2.1.46

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

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Step 2.1.46.1

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

$8 - 24 \sqrt{2} i - 60 + 40 \sqrt{2} i + 30 - 6 \sqrt{2} i + {(i^{2})}^{2} i^{2}$

Step 2.1.46.2

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

$8 - 24 \sqrt{2} i - 60 + 40 \sqrt{2} i + 30 - 6 \sqrt{2} i + {(- 1)}^{2} i^{2}$

Step 2.1.46.3

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

$8 - 24 \sqrt{2} i - 60 + 40 \sqrt{2} i + 30 - 6 \sqrt{2} i + 1 i^{2}$

Step 2.1.47

Multiply $i^{2}$ by $1$ .

$8 - 24 \sqrt{2} i - 60 + 40 \sqrt{2} i + 30 - 6 \sqrt{2} i + i^{2}$

Step 2.1.48

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

$8 - 24 \sqrt{2} i - 60 + 40 \sqrt{2} i + 30 - 6 \sqrt{2} i - 1$

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Subtract $60$ from $8$ .

$- 24 \sqrt{2} i - 52 + 40 \sqrt{2} i + 30 - 6 \sqrt{2} i - 1$

Step 2.2.2

Add $- 24 \sqrt{2} i$ and $40 \sqrt{2} i$ .

$16 \sqrt{2} i - 52 + 30 - 6 \sqrt{2} i - 1$

Step 2.2.3

Subtract $6 \sqrt{2} i$ from $16 \sqrt{2} i$ .

$10 \sqrt{2} i - 52 + 30 - 1$

Step 2.2.4

Simplify the expression.

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Step 2.2.4.1

Add $- 52$ and $30$ .

$10 \sqrt{2} i - 22 - 1$

Step 2.2.4.2

Subtract $1$ from $- 22$ .

$10 \sqrt{2} i - 23$

Step 2.2.4.3

Reorder $10 \sqrt{2} i$ and $- 23$ .

$- 23 + 10 \sqrt{2} i$

Step 3

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 4

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 5

Substitute the actual values of $a = - 23$ and $b = 10 \sqrt{2}$ .

$| z | = \sqrt{{(10 \sqrt{2})}^{2} + {(- 23)}^{2}}$

Step 6

Find $| z |$ .

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Step 6.1

Simplify the expression.

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Step 6.1.1

Apply the product rule to $10 \sqrt{2}$ .

$| z | = \sqrt{10^{2} {\sqrt{2}}^{2} + {(- 23)}^{2}}$

Step 6.1.2

Raise $10$ to the power of $2$ .

$| z | = \sqrt{100 {\sqrt{2}}^{2} + {(- 23)}^{2}}$

Step 6.2

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

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Step 6.2.1

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

$| z | = \sqrt{100 {(2^{\frac{1}{2}})}^{2} + {(- 23)}^{2}}$

Step 6.2.2

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

$| z | = \sqrt{100 \cdot 2^{\frac{1}{2} \cdot 2} + {(- 23)}^{2}}$

Step 6.2.3

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

$| z | = \sqrt{100 \cdot 2^{\frac{2}{2}} + {(- 23)}^{2}}$

Step 6.2.4

Cancel the common factor of $2$ .

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Step 6.2.4.1

Cancel the common factor.

$| z | = \sqrt{100 \cdot 2^{\frac{2}{2}} + {(- 23)}^{2}}$

Step 6.2.4.2

Rewrite the expression.

$| z | = \sqrt{100 \cdot 2 + {(- 23)}^{2}}$

Step 6.2.5

Evaluate the exponent.

$| z | = \sqrt{100 \cdot 2 + {(- 23)}^{2}}$

Step 6.3

Simplify the expression.

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Step 6.3.1

Multiply $100$ by $2$ .

$| z | = \sqrt{200 + {(- 23)}^{2}}$

Step 6.3.2

Raise $- 23$ to the power of $2$ .

$| z | = \sqrt{200 + 529}$

Step 6.3.3

Add $200$ and $529$ .

$| z | = \sqrt{729}$

Step 6.3.4

Rewrite $729$ as $27^{2}$ .

$| z | = \sqrt{27^{2}}$

Step 6.3.5

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

$| z | = 27$

Step 7

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

$θ = \arctan (\frac{10 \sqrt{2}}{- 23})$

Step 8

Since inverse tangent of $\frac{10 \sqrt{2}}{- 23}$ produces an angle in the second quadrant, the value of the angle is $2.59030705$ .

$θ = 2.59030705$

Step 9

Substitute the values of $θ = 2.59030705$ and $| z | = 27$ .

$27 (\cos (2.59030705) + i \sin (2.59030705))$