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

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

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

Step 2.1.2

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

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

Step 2.1.3

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

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

Step 2.1.4

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

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Step 2.1.4.1

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 2.1.4.2

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 2.1.4.3

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 2.1.4.4

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

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Step 2.1.4.4.1

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 2.1.4.4.2

Cancel the common factors.

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Step 2.1.4.4.2.1

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 2.1.4.4.2.2

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 2.1.4.4.2.3

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 2.1.4.4.2.4

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 2.1.5

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 2.1.6

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

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

Step 2.1.7

Multiply ${(- 1)}^{3}$ by $- 1$ by adding the exponents.

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Step 2.1.7.1

Move $- 1$ .

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

Step 2.1.7.2

Multiply $- 1$ by ${(- 1)}^{3}$ .

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Step 2.1.7.2.1

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

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

Step 2.1.7.2.2

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

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

Step 2.1.7.3

Add $1$ and $3$ .

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

Step 2.1.8

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

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

Step 2.1.9

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

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

Step 2.1.10

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 2.1.11

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 2.1.12

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

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Step 2.1.12.1

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 2.1.12.2

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 2.1.13

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 2.1.14

Multiply $3$ by $4$ .

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

Step 2.1.15

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

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

Step 2.1.16

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

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

Step 2.1.17

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

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

Step 2.1.18

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

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Step 2.1.18.1

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 2.1.18.2

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 2.1.18.3

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 2.1.18.4

Cancel the common factor of $2$ .

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Step 2.1.18.4.1

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 2.1.18.4.2

Rewrite the expression.

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

Step 2.1.18.5

Evaluate the exponent.

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

Step 2.1.19

Multiply $6$ by $3$ .

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

Step 2.1.20

Apply the product rule to $- i$ .

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

Step 2.1.21

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

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

Step 2.1.22

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

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

Step 2.1.23

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

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

Step 2.1.24

Multiply $18$ by $- 1$ .

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

Step 2.1.25

Multiply $- 1$ by $4$ .

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

Step 2.1.26

Apply the product rule to $- i$ .

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

Step 2.1.27

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

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

Step 2.1.28

Factor out $i^{2}$ .

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

Step 2.1.29

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

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

Step 2.1.30

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

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

Step 2.1.31

Multiply $- 1$ by $- 1$ .

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

Step 2.1.32

Multiply $i$ by $1$ .

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

Step 2.1.33

Apply the product rule to $- i$ .

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

Step 2.1.34

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

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

Step 2.1.35

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

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

Step 2.1.36

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

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Step 2.1.36.1

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

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

Step 2.1.36.2

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

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

Step 2.1.36.3

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

Subtract $4 \sqrt{3} i$ from $12 \sqrt{3} i$ .

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

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 $- 8$ .

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