Precalculus Examples

${(- 2 - 2 i)}^{4}$

Step 1

Use the Binomial Theorem.

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

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Raise $- 2$ to the power of $4$ .

$16 + 4 {(- 2)}^{3} (- 2 i) + 6 {(- 2)}^{2} {(- 2 i)}^{2} + 4 \cdot - 2 {(- 2 i)}^{3} + {(- 2 i)}^{4}$

Step 2.1.2

Multiply ${(- 2)}^{3}$ by $- 2$ by adding the exponents.

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Step 2.1.2.1

Move $- 2$ .

$16 + 4 (- 2 {(- 2)}^{3}) i + 6 {(- 2)}^{2} {(- 2 i)}^{2} + 4 \cdot - 2 {(- 2 i)}^{3} + {(- 2 i)}^{4}$

Step 2.1.2.2

Multiply $- 2$ by ${(- 2)}^{3}$ .

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Step 2.1.2.2.1

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

$16 + 4 ({(- 2)}^{1} {(- 2)}^{3}) i + 6 {(- 2)}^{2} {(- 2 i)}^{2} + 4 \cdot - 2 {(- 2 i)}^{3} + {(- 2 i)}^{4}$

Step 2.1.2.2.2

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

$16 + 4 {(- 2)}^{1 + 3} i + 6 {(- 2)}^{2} {(- 2 i)}^{2} + 4 \cdot - 2 {(- 2 i)}^{3} + {(- 2 i)}^{4}$

Step 2.1.2.3

Add $1$ and $3$ .

$16 + 4 {(- 2)}^{4} i + 6 {(- 2)}^{2} {(- 2 i)}^{2} + 4 \cdot - 2 {(- 2 i)}^{3} + {(- 2 i)}^{4}$

Step 2.1.3

Raise $- 2$ to the power of $4$ .

$16 + 4 \cdot 16 i + 6 {(- 2)}^{2} {(- 2 i)}^{2} + 4 \cdot - 2 {(- 2 i)}^{3} + {(- 2 i)}^{4}$

Step 2.1.4

Multiply $4$ by $16$ .

$16 + 64 i + 6 {(- 2)}^{2} {(- 2 i)}^{2} + 4 \cdot - 2 {(- 2 i)}^{3} + {(- 2 i)}^{4}$

Step 2.1.5

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

$16 + 64 i + 6 \cdot 4 {(- 2 i)}^{2} + 4 \cdot - 2 {(- 2 i)}^{3} + {(- 2 i)}^{4}$

Step 2.1.6

Multiply $6$ by $4$ .

$16 + 64 i + 24 {(- 2 i)}^{2} + 4 \cdot - 2 {(- 2 i)}^{3} + {(- 2 i)}^{4}$

Step 2.1.7

Apply the product rule to $- 2 i$ .

$16 + 64 i + 24 ({(- 2)}^{2} i^{2}) + 4 \cdot - 2 {(- 2 i)}^{3} + {(- 2 i)}^{4}$

Step 2.1.8

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

$16 + 64 i + 24 (4 i^{2}) + 4 \cdot - 2 {(- 2 i)}^{3} + {(- 2 i)}^{4}$

Step 2.1.9

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

$16 + 64 i + 24 (4 \cdot - 1) + 4 \cdot - 2 {(- 2 i)}^{3} + {(- 2 i)}^{4}$

Step 2.1.10

Multiply $24 (4 \cdot - 1)$ .

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Step 2.1.10.1

Multiply $4$ by $- 1$ .

$16 + 64 i + 24 \cdot - 4 + 4 \cdot - 2 {(- 2 i)}^{3} + {(- 2 i)}^{4}$

Step 2.1.10.2

Multiply $24$ by $- 4$ .

$16 + 64 i - 96 + 4 \cdot - 2 {(- 2 i)}^{3} + {(- 2 i)}^{4}$

Step 2.1.11

Multiply $4$ by $- 2$ .

$16 + 64 i - 96 - 8 {(- 2 i)}^{3} + {(- 2 i)}^{4}$

Step 2.1.12

Apply the product rule to $- 2 i$ .

$16 + 64 i - 96 - 8 ({(- 2)}^{3} i^{3}) + {(- 2 i)}^{4}$

Step 2.1.13

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

$16 + 64 i - 96 - 8 (- 8 i^{3}) + {(- 2 i)}^{4}$

Step 2.1.14

Factor out $i^{2}$ .

$16 + 64 i - 96 - 8 (- 8 (i^{2} \cdot i)) + {(- 2 i)}^{4}$

Step 2.1.15

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

$16 + 64 i - 96 - 8 (- 8 (- 1 \cdot i)) + {(- 2 i)}^{4}$

Step 2.1.16

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

$16 + 64 i - 96 - 8 (- 8 (- i)) + {(- 2 i)}^{4}$

Step 2.1.17

Multiply $- 1$ by $- 8$ .

$16 + 64 i - 96 - 8 (8 i) + {(- 2 i)}^{4}$

Step 2.1.18

Multiply $8$ by $- 8$ .

$16 + 64 i - 96 - 64 i + {(- 2 i)}^{4}$

Step 2.1.19

Apply the product rule to $- 2 i$ .

$16 + 64 i - 96 - 64 i + {(- 2)}^{4} i^{4}$

Step 2.1.20

Raise $- 2$ to the power of $4$ .

$16 + 64 i - 96 - 64 i + 16 i^{4}$

Step 2.1.21

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

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Step 2.1.21.1

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

$16 + 64 i - 96 - 64 i + 16 {(i^{2})}^{2}$

Step 2.1.21.2

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

$16 + 64 i - 96 - 64 i + 16 {(- 1)}^{2}$

Step 2.1.21.3

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

$16 + 64 i - 96 - 64 i + 16 \cdot 1$

Step 2.1.22

Multiply $16$ by $1$ .

$16 + 64 i - 96 - 64 i + 16$

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Subtract $96$ from $16$ .

$- 80 + 64 i - 64 i + 16$

Step 2.2.2

Add $- 80$ and $16$ .

$- 64 + 64 i - 64 i$

Step 2.2.3

Subtract $64 i$ from $64 i$ .

$- 64 + 0$

Step 2.2.4

Add $- 64$ and $0$ .

$- 64$

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 $a = - 64$ and $b = 0$ .

$| z | = \sqrt{0^{2} + {(- 64)}^{2}}$

Step 6

Find $| z |$ .

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Step 6.1

Raising $0$ to any positive power yields $0$ .

$| z | = \sqrt{0 + {(- 64)}^{2}}$

Step 6.2

Raise $- 64$ to the power of $2$ .

$| z | = \sqrt{0 + 4096}$

Step 6.3

Add $0$ and $4096$ .

$| z | = \sqrt{4096}$

Step 6.4

Rewrite $4096$ as $64^{2}$ .

$| z | = \sqrt{64^{2}}$

Step 6.5

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

$| z | = 64$

Step 7

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

$θ = \arctan (\frac{0}{- 64})$

Step 8

Since inverse tangent of $\frac{0}{- 64}$ produces an angle in the second quadrant, the value of the angle is $π$ .

$θ = π$

Step 9

Substitute the values of $θ = π$ and $| z | = 64$ .

$64 (\cos (π) + i \sin (π))$