Trigonometry Examples

${(1 - \sqrt{3} i)}^{5}$

Step 1

Use the Binomial Theorem.

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

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

One to any power is one.

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

Step 2.1.2

One to any power is one.

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

Step 2.1.3

Multiply $5$ by $1$ .

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

Step 2.1.4

Multiply $- 1$ by $5$ .

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

Step 2.1.5

One to any power is one.

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

Step 2.1.6

Multiply $10$ by $1$ .

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

Step 2.1.7

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

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Step 2.1.7.1

Apply the product rule to $- \sqrt{3} i$ .

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

Step 2.1.7.2

Apply the product rule to $- \sqrt{3}$ .

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

Step 2.1.8

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

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

Step 2.1.9

Multiply ${\sqrt{3}}^{2}$ by $1$ .

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

Step 2.1.10

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

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Step 2.1.10.1

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

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

Step 2.1.10.2

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

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

Step 2.1.10.3

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

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

Step 2.1.10.4

Cancel the common factor of $2$ .

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Step 2.1.10.4.1

Cancel the common factor.

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

Step 2.1.10.4.2

Rewrite the expression.

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

Step 2.1.10.5

Evaluate the exponent.

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

Step 2.1.11

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

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

Step 2.1.12

Multiply $10 (3 \cdot - 1)$ .

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Step 2.1.12.1

Multiply $3$ by $- 1$ .

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

Step 2.1.12.2

Multiply $10$ by $- 3$ .

$1 - 5 \sqrt{3} i - 30 + 10 \cdot 1^{2} {(- \sqrt{3} i)}^{3} + 5 \cdot 1 {(- \sqrt{3} i)}^{4} + {(- \sqrt{3} i)}^{5}$

Step 2.1.13

One to any power is one.

$1 - 5 \sqrt{3} i - 30 + 10 \cdot 1 {(- \sqrt{3} i)}^{3} + 5 \cdot 1 {(- \sqrt{3} i)}^{4} + {(- \sqrt{3} i)}^{5}$

Step 2.1.14

Multiply $10$ by $1$ .

$1 - 5 \sqrt{3} i - 30 + 10 {(- \sqrt{3} i)}^{3} + 5 \cdot 1 {(- \sqrt{3} i)}^{4} + {(- \sqrt{3} i)}^{5}$

Step 2.1.15

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

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Step 2.1.15.1

Apply the product rule to $- \sqrt{3} i$ .

$1 - 5 \sqrt{3} i - 30 + 10 ({(- \sqrt{3})}^{3} i^{3}) + 5 \cdot 1 {(- \sqrt{3} i)}^{4} + {(- \sqrt{3} i)}^{5}$

Step 2.1.15.2

Apply the product rule to $- \sqrt{3}$ .

$1 - 5 \sqrt{3} i - 30 + 10 ({(- 1)}^{3} {\sqrt{3}}^{3} i^{3}) + 5 \cdot 1 {(- \sqrt{3} i)}^{4} + {(- \sqrt{3} i)}^{5}$

Step 2.1.16

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

$1 - 5 \sqrt{3} i - 30 + 10 (- {\sqrt{3}}^{3} i^{3}) + 5 \cdot 1 {(- \sqrt{3} i)}^{4} + {(- \sqrt{3} i)}^{5}$

Step 2.1.17

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

$1 - 5 \sqrt{3} i - 30 + 10 (- \sqrt{3^{3}} i^{3}) + 5 \cdot 1 {(- \sqrt{3} i)}^{4} + {(- \sqrt{3} i)}^{5}$

Step 2.1.18

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

$1 - 5 \sqrt{3} i - 30 + 10 (- \sqrt{27} i^{3}) + 5 \cdot 1 {(- \sqrt{3} i)}^{4} + {(- \sqrt{3} i)}^{5}$

Step 2.1.19

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 2.1.19.1

Factor $9$ out of $27$ .

$1 - 5 \sqrt{3} i - 30 + 10 (- \sqrt{9 (3)} i^{3}) + 5 \cdot 1 {(- \sqrt{3} i)}^{4} + {(- \sqrt{3} i)}^{5}$

Step 2.1.19.2

Rewrite $9$ as $3^{2}$ .

$1 - 5 \sqrt{3} i - 30 + 10 (- \sqrt{3^{2} \cdot 3} i^{3}) + 5 \cdot 1 {(- \sqrt{3} i)}^{4} + {(- \sqrt{3} i)}^{5}$

Step 2.1.20

Pull terms out from under the radical.

$1 - 5 \sqrt{3} i - 30 + 10 (- (3 \sqrt{3}) i^{3}) + 5 \cdot 1 {(- \sqrt{3} i)}^{4} + {(- \sqrt{3} i)}^{5}$

Step 2.1.21

Multiply $3$ by $- 1$ .

$1 - 5 \sqrt{3} i - 30 + 10 (- 3 \sqrt{3} i^{3}) + 5 \cdot 1 {(- \sqrt{3} i)}^{4} + {(- \sqrt{3} i)}^{5}$

Step 2.1.22

Factor out $i^{2}$ .

$1 - 5 \sqrt{3} i - 30 + 10 (- 3 \sqrt{3} (i^{2} \cdot i)) + 5 \cdot 1 {(- \sqrt{3} i)}^{4} + {(- \sqrt{3} i)}^{5}$

Step 2.1.23

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

$1 - 5 \sqrt{3} i - 30 + 10 (- 3 \sqrt{3} (- 1 \cdot i)) + 5 \cdot 1 {(- \sqrt{3} i)}^{4} + {(- \sqrt{3} i)}^{5}$

Step 2.1.24

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

$1 - 5 \sqrt{3} i - 30 + 10 (- 3 \sqrt{3} (- i)) + 5 \cdot 1 {(- \sqrt{3} i)}^{4} + {(- \sqrt{3} i)}^{5}$

Step 2.1.25

Multiply $- 1$ by $- 3$ .

$1 - 5 \sqrt{3} i - 30 + 10 (3 \sqrt{3} i) + 5 \cdot 1 {(- \sqrt{3} i)}^{4} + {(- \sqrt{3} i)}^{5}$

Step 2.1.26

Multiply $3$ by $10$ .

$1 - 5 \sqrt{3} i - 30 + 30 (\sqrt{3} i) + 5 \cdot 1 {(- \sqrt{3} i)}^{4} + {(- \sqrt{3} i)}^{5}$

Step 2.1.27

Multiply $5$ by $1$ .

$1 - 5 \sqrt{3} i - 30 + 30 \sqrt{3} i + 5 {(- \sqrt{3} i)}^{4} + {(- \sqrt{3} i)}^{5}$

Step 2.1.28

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

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Step 2.1.28.1

Apply the product rule to $- \sqrt{3} i$ .

$1 - 5 \sqrt{3} i - 30 + 30 \sqrt{3} i + 5 ({(- \sqrt{3})}^{4} i^{4}) + {(- \sqrt{3} i)}^{5}$

Step 2.1.28.2

Apply the product rule to $- \sqrt{3}$ .

$1 - 5 \sqrt{3} i - 30 + 30 \sqrt{3} i + 5 ({(- 1)}^{4} {\sqrt{3}}^{4} i^{4}) + {(- \sqrt{3} i)}^{5}$

Step 2.1.29

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

$1 - 5 \sqrt{3} i - 30 + 30 \sqrt{3} i + 5 (1 {\sqrt{3}}^{4} i^{4}) + {(- \sqrt{3} i)}^{5}$

Step 2.1.30

Multiply ${\sqrt{3}}^{4}$ by $1$ .

$1 - 5 \sqrt{3} i - 30 + 30 \sqrt{3} i + 5 ({\sqrt{3}}^{4} i^{4}) + {(- \sqrt{3} i)}^{5}$

Step 2.1.31

Rewrite ${\sqrt{3}}^{4}$ as $3^{2}$ .

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Step 2.1.31.1

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

$1 - 5 \sqrt{3} i - 30 + 30 \sqrt{3} i + 5 ({(3^{\frac{1}{2}})}^{4} i^{4}) + {(- \sqrt{3} i)}^{5}$

Step 2.1.31.2

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

$1 - 5 \sqrt{3} i - 30 + 30 \sqrt{3} i + 5 (3^{\frac{1}{2} \cdot 4} i^{4}) + {(- \sqrt{3} i)}^{5}$

Step 2.1.31.3

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

$1 - 5 \sqrt{3} i - 30 + 30 \sqrt{3} i + 5 (3^{\frac{4}{2}} i^{4}) + {(- \sqrt{3} i)}^{5}$

Step 2.1.31.4

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

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Step 2.1.31.4.1

Factor $2$ out of $4$ .

$1 - 5 \sqrt{3} i - 30 + 30 \sqrt{3} i + 5 (3^{\frac{2 \cdot 2}{2}} i^{4}) + {(- \sqrt{3} i)}^{5}$

Step 2.1.31.4.2

Cancel the common factors.

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Step 2.1.31.4.2.1

Factor $2$ out of $2$ .

$1 - 5 \sqrt{3} i - 30 + 30 \sqrt{3} i + 5 (3^{\frac{2 \cdot 2}{2 (1)}} i^{4}) + {(- \sqrt{3} i)}^{5}$

Step 2.1.31.4.2.2

Cancel the common factor.

$1 - 5 \sqrt{3} i - 30 + 30 \sqrt{3} i + 5 (3^{\frac{2 \cdot 2}{2 \cdot 1}} i^{4}) + {(- \sqrt{3} i)}^{5}$

Step 2.1.31.4.2.3

Rewrite the expression.

$1 - 5 \sqrt{3} i - 30 + 30 \sqrt{3} i + 5 (3^{\frac{2}{1}} i^{4}) + {(- \sqrt{3} i)}^{5}$

Step 2.1.31.4.2.4

Divide $2$ by $1$ .

$1 - 5 \sqrt{3} i - 30 + 30 \sqrt{3} i + 5 (3^{2} i^{4}) + {(- \sqrt{3} i)}^{5}$

Step 2.1.32

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

$1 - 5 \sqrt{3} i - 30 + 30 \sqrt{3} i + 5 (9 i^{4}) + {(- \sqrt{3} i)}^{5}$

Step 2.1.33

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

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Step 2.1.33.1

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

$1 - 5 \sqrt{3} i - 30 + 30 \sqrt{3} i + 5 (9 {(i^{2})}^{2}) + {(- \sqrt{3} i)}^{5}$

Step 2.1.33.2

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

$1 - 5 \sqrt{3} i - 30 + 30 \sqrt{3} i + 5 (9 {(- 1)}^{2}) + {(- \sqrt{3} i)}^{5}$

Step 2.1.33.3

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

$1 - 5 \sqrt{3} i - 30 + 30 \sqrt{3} i + 5 (9 \cdot 1) + {(- \sqrt{3} i)}^{5}$

Step 2.1.34

Multiply $5 (9 \cdot 1)$ .

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Step 2.1.34.1

Multiply $9$ by $1$ .

$1 - 5 \sqrt{3} i - 30 + 30 \sqrt{3} i + 5 \cdot 9 + {(- \sqrt{3} i)}^{5}$

Step 2.1.34.2

Multiply $5$ by $9$ .

$1 - 5 \sqrt{3} i - 30 + 30 \sqrt{3} i + 45 + {(- \sqrt{3} i)}^{5}$

Step 2.1.35

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

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Step 2.1.35.1

Apply the product rule to $- \sqrt{3} i$ .

$1 - 5 \sqrt{3} i - 30 + 30 \sqrt{3} i + 45 + {(- \sqrt{3})}^{5} i^{5}$

Step 2.1.35.2

Apply the product rule to $- \sqrt{3}$ .

$1 - 5 \sqrt{3} i - 30 + 30 \sqrt{3} i + 45 + {(- 1)}^{5} {\sqrt{3}}^{5} i^{5}$

Step 2.1.36

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

$1 - 5 \sqrt{3} i - 30 + 30 \sqrt{3} i + 45 - {\sqrt{3}}^{5} i^{5}$

Step 2.1.37

Rewrite ${\sqrt{3}}^{5}$ as $\sqrt{3^{5}}$ .

$1 - 5 \sqrt{3} i - 30 + 30 \sqrt{3} i + 45 - \sqrt{3^{5}} i^{5}$

Step 2.1.38

Raise $3$ to the power of $5$ .

$1 - 5 \sqrt{3} i - 30 + 30 \sqrt{3} i + 45 - \sqrt{243} i^{5}$

Step 2.1.39

Rewrite $243$ as $9^{2} \cdot 3$ .

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Step 2.1.39.1

Factor $81$ out of $243$ .

$1 - 5 \sqrt{3} i - 30 + 30 \sqrt{3} i + 45 - \sqrt{81 (3)} i^{5}$

Step 2.1.39.2

Rewrite $81$ as $9^{2}$ .

$1 - 5 \sqrt{3} i - 30 + 30 \sqrt{3} i + 45 - \sqrt{9^{2} \cdot 3} i^{5}$

Step 2.1.40

Pull terms out from under the radical.

$1 - 5 \sqrt{3} i - 30 + 30 \sqrt{3} i + 45 - (9 \sqrt{3}) i^{5}$

Step 2.1.41

Multiply $9$ by $- 1$ .

$1 - 5 \sqrt{3} i - 30 + 30 \sqrt{3} i + 45 - 9 \sqrt{3} i^{5}$

Step 2.1.42

Factor out $i^{4}$ .

$1 - 5 \sqrt{3} i - 30 + 30 \sqrt{3} i + 45 - 9 \sqrt{3} (i^{4} i)$

Step 2.1.43

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

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Step 2.1.43.1

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

$1 - 5 \sqrt{3} i - 30 + 30 \sqrt{3} i + 45 - 9 \sqrt{3} ({(i^{2})}^{2} i)$

Step 2.1.43.2

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

$1 - 5 \sqrt{3} i - 30 + 30 \sqrt{3} i + 45 - 9 \sqrt{3} ({(- 1)}^{2} i)$

Step 2.1.43.3

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

$1 - 5 \sqrt{3} i - 30 + 30 \sqrt{3} i + 45 - 9 \sqrt{3} (1 i)$

Step 2.1.44

Multiply $i$ by $1$ .

$1 - 5 \sqrt{3} i - 30 + 30 \sqrt{3} i + 45 - 9 \sqrt{3} i$

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Subtract $30$ from $1$ .

$- 5 \sqrt{3} i - 29 + 30 \sqrt{3} i + 45 - 9 \sqrt{3} i$

Step 2.2.2

Add $- 5 \sqrt{3} i$ and $30 \sqrt{3} i$ .

$25 \sqrt{3} i - 29 + 45 - 9 \sqrt{3} i$

Step 2.2.3

Subtract $9 \sqrt{3} i$ from $25 \sqrt{3} i$ .

$16 \sqrt{3} i - 29 + 45$

Step 2.2.4

Simplify the expression.

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Step 2.2.4.1

Add $- 29$ and $45$ .

$16 \sqrt{3} i + 16$

Step 2.2.4.2

Reorder $16 \sqrt{3} i$ and $16$ .

$16 + 16 \sqrt{3} i$