Trigonometry Examples

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

Step 1

Use the Binomial Theorem.

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

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

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

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Step 2.1.1.1

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

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

Step 2.1.1.2

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

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

Step 2.1.1.3

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

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

Step 2.1.1.4

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

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Step 2.1.1.4.1

Factor $2$ out of $4$ .

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

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 $2$ .

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

Step 2.1.1.4.2.2

Cancel the common factor.

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

Step 2.1.1.4.2.3

Rewrite the expression.

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

Step 2.1.1.4.2.4

Divide $2$ by $1$ .

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

Step 2.1.2

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

$9 + 4 {\sqrt{3}}^{3} (- i) + 6 {\sqrt{3}}^{2} {(- i)}^{2} + 4 \sqrt{3} {(- i)}^{3} + {(- i)}^{4}$

Step 2.1.3

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

$9 + 4 \sqrt{3^{3}} (- i) + 6 {\sqrt{3}}^{2} {(- i)}^{2} + 4 \sqrt{3} {(- i)}^{3} + {(- i)}^{4}$

Step 2.1.4

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

$9 + 4 \sqrt{27} (- i) + 6 {\sqrt{3}}^{2} {(- i)}^{2} + 4 \sqrt{3} {(- i)}^{3} + {(- i)}^{4}$

Step 2.1.5

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

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Step 2.1.5.1

Factor $9$ out of $27$ .

$9 + 4 \sqrt{9 (3)} (- i) + 6 {\sqrt{3}}^{2} {(- i)}^{2} + 4 \sqrt{3} {(- i)}^{3} + {(- i)}^{4}$

Step 2.1.5.2

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

$9 + 4 \sqrt{3^{2} \cdot 3} (- i) + 6 {\sqrt{3}}^{2} {(- i)}^{2} + 4 \sqrt{3} {(- i)}^{3} + {(- i)}^{4}$

Step 2.1.6

Pull terms out from under the radical.

$9 + 4 (3 \sqrt{3}) (- i) + 6 {\sqrt{3}}^{2} {(- i)}^{2} + 4 \sqrt{3} {(- i)}^{3} + {(- i)}^{4}$

Step 2.1.7

Multiply $3$ by $4$ .

$9 + 12 \sqrt{3} (- i) + 6 {\sqrt{3}}^{2} {(- i)}^{2} + 4 \sqrt{3} {(- i)}^{3} + {(- i)}^{4}$

Step 2.1.8

Multiply $- 1$ by $12$ .

$9 - 12 \sqrt{3} i + 6 {\sqrt{3}}^{2} {(- i)}^{2} + 4 \sqrt{3} {(- i)}^{3} + {(- i)}^{4}$

Step 2.1.9

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

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Step 2.1.9.1

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

$9 - 12 \sqrt{3} i + 6 {(3^{\frac{1}{2}})}^{2} {(- i)}^{2} + 4 \sqrt{3} {(- i)}^{3} + {(- i)}^{4}$

Step 2.1.9.2

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

$9 - 12 \sqrt{3} i + 6 \cdot 3^{\frac{1}{2} \cdot 2} {(- i)}^{2} + 4 \sqrt{3} {(- i)}^{3} + {(- i)}^{4}$

Step 2.1.9.3

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

$9 - 12 \sqrt{3} i + 6 \cdot 3^{\frac{2}{2}} {(- i)}^{2} + 4 \sqrt{3} {(- i)}^{3} + {(- i)}^{4}$

Step 2.1.9.4

Cancel the common factor of $2$ .

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Step 2.1.9.4.1

Cancel the common factor.

$9 - 12 \sqrt{3} i + 6 \cdot 3^{\frac{2}{2}} {(- i)}^{2} + 4 \sqrt{3} {(- i)}^{3} + {(- i)}^{4}$

Step 2.1.9.4.2

Rewrite the expression.

$9 - 12 \sqrt{3} i + 6 \cdot 3^{1} {(- i)}^{2} + 4 \sqrt{3} {(- i)}^{3} + {(- i)}^{4}$

Step 2.1.9.5

Evaluate the exponent.

$9 - 12 \sqrt{3} i + 6 \cdot 3 {(- i)}^{2} + 4 \sqrt{3} {(- i)}^{3} + {(- i)}^{4}$

Step 2.1.10

Multiply $6$ by $3$ .

$9 - 12 \sqrt{3} i + 18 {(- i)}^{2} + 4 \sqrt{3} {(- i)}^{3} + {(- i)}^{4}$

Step 2.1.11

Apply the product rule to $- i$ .

$9 - 12 \sqrt{3} i + 18 ({(- 1)}^{2} i^{2}) + 4 \sqrt{3} {(- i)}^{3} + {(- i)}^{4}$

Step 2.1.12

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

$9 - 12 \sqrt{3} i + 18 (1 i^{2}) + 4 \sqrt{3} {(- i)}^{3} + {(- i)}^{4}$

Step 2.1.13

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

$9 - 12 \sqrt{3} i + 18 i^{2} + 4 \sqrt{3} {(- i)}^{3} + {(- i)}^{4}$

Step 2.1.14

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

$9 - 12 \sqrt{3} i + 18 \cdot - 1 + 4 \sqrt{3} {(- i)}^{3} + {(- i)}^{4}$

Step 2.1.15

Multiply $18$ by $- 1$ .

$9 - 12 \sqrt{3} i - 18 + 4 \sqrt{3} {(- i)}^{3} + {(- i)}^{4}$

Step 2.1.16

Apply the product rule to $- i$ .

$9 - 12 \sqrt{3} i - 18 + 4 \sqrt{3} ({(- 1)}^{3} i^{3}) + {(- i)}^{4}$

Step 2.1.17

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

$9 - 12 \sqrt{3} i - 18 + 4 \sqrt{3} (- i^{3}) + {(- i)}^{4}$

Step 2.1.18

Factor out $i^{2}$ .

$9 - 12 \sqrt{3} i - 18 + 4 \sqrt{3} (- (i^{2} \cdot i)) + {(- i)}^{4}$

Step 2.1.19

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

$9 - 12 \sqrt{3} i - 18 + 4 \sqrt{3} (- (- 1 \cdot i)) + {(- i)}^{4}$

Step 2.1.20

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

$9 - 12 \sqrt{3} i - 18 + 4 \sqrt{3} (- - i) + {(- i)}^{4}$

Step 2.1.21

Multiply $- 1$ by $- 1$ .

$9 - 12 \sqrt{3} i - 18 + 4 \sqrt{3} (1 i) + {(- i)}^{4}$

Step 2.1.22

Multiply $i$ by $1$ .

$9 - 12 \sqrt{3} i - 18 + 4 \sqrt{3} i + {(- i)}^{4}$

Step 2.1.23

Apply the product rule to $- i$ .

$9 - 12 \sqrt{3} i - 18 + 4 \sqrt{3} i + {(- 1)}^{4} i^{4}$

Step 2.1.24

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

$9 - 12 \sqrt{3} i - 18 + 4 \sqrt{3} i + 1 i^{4}$

Step 2.1.25

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

$9 - 12 \sqrt{3} i - 18 + 4 \sqrt{3} i + i^{4}$

Step 2.1.26

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

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Step 2.1.26.1

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

$9 - 12 \sqrt{3} i - 18 + 4 \sqrt{3} i + {(i^{2})}^{2}$

Step 2.1.26.2

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

$9 - 12 \sqrt{3} i - 18 + 4 \sqrt{3} i + {(- 1)}^{2}$

Step 2.1.26.3

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

$9 - 12 \sqrt{3} i - 18 + 4 \sqrt{3} i + 1$

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Subtract $18$ from $9$ .

$- 12 \sqrt{3} i - 9 + 4 \sqrt{3} i + 1$

Step 2.2.2

Add $- 12 \sqrt{3} i$ and $4 \sqrt{3} i$ .

$- 8 \sqrt{3} i - 9 + 1$

Step 2.2.3

Simplify the expression.

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Step 2.2.3.1

Add $- 9$ and $1$ .

$- 8 \sqrt{3} i - 8$

Step 2.2.3.2

Reorder $- 8 \sqrt{3} i$ and $- 8$ .

$- 8 - 8 \sqrt{3} i$