Precalculus Examples

${(2 (\cos (240 °) + i \sin (240 °)))}^{4}$

Step 1

Simplify terms.

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the third quadrant.

${(2 (- \cos (60) + i \sin (240 °)))}^{4}$

Step 1.1.2

The exact value of $\cos (60)$ is $\frac{1}{2}$ .

${(2 (- \frac{1}{2} + i \sin (240 °)))}^{4}$

Step 1.1.3

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the third quadrant.

${(2 (- \frac{1}{2} + i (- \sin (60))))}^{4}$

Step 1.1.4

The exact value of $\sin (60)$ is $\frac{\sqrt{3}}{2}$ .

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

Step 1.1.5

Combine $i$ and $\frac{\sqrt{3}}{2}$ .

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

Step 1.2

Simplify terms.

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Step 1.2.1

Apply the distributive property.

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

Step 1.2.2

Cancel the common factor of $2$ .

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Step 1.2.2.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

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

Step 1.2.2.2

Cancel the common factor.

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

Step 1.2.2.3

Rewrite the expression.

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

Step 1.2.3

Cancel the common factor of $2$ .

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Step 1.2.3.1

Move the leading negative in $- \frac{i \sqrt{3}}{2}$ into the numerator.

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

Step 1.2.3.2

Cancel the common factor.

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

Step 1.2.3.3

Rewrite the expression.

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

Step 2

Use the Binomial Theorem.

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

Step 3

Simplify terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

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

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

Step 3.1.2

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

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Step 3.1.2.1

Move $- 1$ .

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

Step 3.1.2.2

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

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Step 3.1.2.2.1

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

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

Step 3.1.2.2.2

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

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

Step 3.1.2.3

Add $1$ and $3$ .

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

Step 3.1.3

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

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

Step 3.1.4

Multiply $4$ by $1$ .

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

Step 3.1.5

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

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

Step 3.1.6

Multiply $6$ by $1$ .

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

Step 3.1.7

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

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Step 3.1.7.1

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

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

Step 3.1.7.2

Apply the product rule to $- i$ .

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

Step 3.1.8

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

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

Step 3.1.9

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

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

Step 3.1.10

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

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

Step 3.1.11

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

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Step 3.1.11.1

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

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

Step 3.1.11.2

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

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

Step 3.1.11.3

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

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

Step 3.1.11.4

Cancel the common factor of $2$ .

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Step 3.1.11.4.1

Cancel the common factor.

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

Step 3.1.11.4.2

Rewrite the expression.

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

Step 3.1.11.5

Evaluate the exponent.

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

Step 3.1.12

Multiply $6 (- 1 \cdot 3)$ .

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Step 3.1.12.1

Multiply $- 1$ by $3$ .

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

Step 3.1.12.2

Multiply $6$ by $- 3$ .

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

Step 3.1.13

Multiply $4$ by $- 1$ .

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

Step 3.1.14

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

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Step 3.1.14.1

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

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

Step 3.1.14.2

Apply the product rule to $- i$ .

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

Step 3.1.15

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

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

Step 3.1.16

Factor out $i^{2}$ .

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

Step 3.1.17

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

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

Step 3.1.18

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

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

Step 3.1.19

Multiply $- 1$ by $- 1$ .

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

Step 3.1.20

Multiply $i$ by $1$ .

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

Step 3.1.21

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

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

Step 3.1.22

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

$1 + 4 i \sqrt{3} - 18 - 4 (i \sqrt{27}) + {(- i \sqrt{3})}^{4}$

Step 3.1.23

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

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Step 3.1.23.1

Factor $9$ out of $27$ .

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

Step 3.1.23.2

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

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

Step 3.1.24

Pull terms out from under the radical.

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

Step 3.1.25

Move $3$ to the left of $i$ .

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

Step 3.1.26

Multiply $3$ by $- 4$ .

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

Step 3.1.27

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

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Step 3.1.27.1

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

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

Step 3.1.27.2

Apply the product rule to $- i$ .

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

Step 3.1.28

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

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

Step 3.1.29

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

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

Step 3.1.30

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

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Step 3.1.30.1

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

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

Step 3.1.30.2

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

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

Step 3.1.30.3

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

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

Step 3.1.31

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

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

Step 3.1.32

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

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Step 3.1.32.1

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

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

Step 3.1.32.2

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

$1 + 4 i \sqrt{3} - 18 - 12 i \sqrt{3} + 3^{\frac{1}{2} \cdot 4}$

Step 3.1.32.3

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

$1 + 4 i \sqrt{3} - 18 - 12 i \sqrt{3} + 3^{\frac{4}{2}}$

Step 3.1.32.4

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

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Step 3.1.32.4.1

Factor $2$ out of $4$ .

$1 + 4 i \sqrt{3} - 18 - 12 i \sqrt{3} + 3^{\frac{2 \cdot 2}{2}}$

Step 3.1.32.4.2

Cancel the common factors.

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Step 3.1.32.4.2.1

Factor $2$ out of $2$ .

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

Step 3.1.32.4.2.2

Cancel the common factor.

$1 + 4 i \sqrt{3} - 18 - 12 i \sqrt{3} + 3^{\frac{2 \cdot 2}{2 \cdot 1}}$

Step 3.1.32.4.2.3

Rewrite the expression.

$1 + 4 i \sqrt{3} - 18 - 12 i \sqrt{3} + 3^{\frac{2}{1}}$

Step 3.1.32.4.2.4

Divide $2$ by $1$ .

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

Step 3.1.33

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

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

Step 3.2

Simplify by adding terms.

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Step 3.2.1

Subtract $18$ from $1$ .

$4 i \sqrt{3} - 17 - 12 i \sqrt{3} + 9$

Step 3.2.2

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

$- 8 i \sqrt{3} - 17 + 9$

Step 3.2.3

Simplify the expression.

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Step 3.2.3.1

Add $- 17$ and $9$ .

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

Step 3.2.3.2

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

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