Precalculus Examples

${(4 + 4 i)}^{5}$

Step 1

Use the Binomial Theorem.

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

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Raise $4$ to the power of $5$ .

$1024 + 5 \cdot 4^{4} (4 i) + 10 \cdot 4^{3} {(4 i)}^{2} + 10 \cdot 4^{2} {(4 i)}^{3} + 5 \cdot 4 {(4 i)}^{4} + {(4 i)}^{5}$

Step 2.1.2

Multiply $4^{4}$ by $4$ by adding the exponents.

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Step 2.1.2.1

Move $4$ .

$1024 + 5 \cdot (4 \cdot 4^{4}) i + 10 \cdot 4^{3} {(4 i)}^{2} + 10 \cdot 4^{2} {(4 i)}^{3} + 5 \cdot 4 {(4 i)}^{4} + {(4 i)}^{5}$

Step 2.1.2.2

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

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Step 2.1.2.2.1

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

$1024 + 5 \cdot (4^{1} \cdot 4^{4}) i + 10 \cdot 4^{3} {(4 i)}^{2} + 10 \cdot 4^{2} {(4 i)}^{3} + 5 \cdot 4 {(4 i)}^{4} + {(4 i)}^{5}$

Step 2.1.2.2.2

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

$1024 + 5 \cdot 4^{1 + 4} i + 10 \cdot 4^{3} {(4 i)}^{2} + 10 \cdot 4^{2} {(4 i)}^{3} + 5 \cdot 4 {(4 i)}^{4} + {(4 i)}^{5}$

Step 2.1.2.3

Add $1$ and $4$ .

$1024 + 5 \cdot 4^{5} i + 10 \cdot 4^{3} {(4 i)}^{2} + 10 \cdot 4^{2} {(4 i)}^{3} + 5 \cdot 4 {(4 i)}^{4} + {(4 i)}^{5}$

Step 2.1.3

Raise $4$ to the power of $5$ .

$1024 + 5 \cdot 1024 i + 10 \cdot 4^{3} {(4 i)}^{2} + 10 \cdot 4^{2} {(4 i)}^{3} + 5 \cdot 4 {(4 i)}^{4} + {(4 i)}^{5}$

Step 2.1.4

Multiply $5$ by $1024$ .

$1024 + 5120 i + 10 \cdot 4^{3} {(4 i)}^{2} + 10 \cdot 4^{2} {(4 i)}^{3} + 5 \cdot 4 {(4 i)}^{4} + {(4 i)}^{5}$

Step 2.1.5

Raise $4$ to the power of $3$ .

$1024 + 5120 i + 10 \cdot 64 {(4 i)}^{2} + 10 \cdot 4^{2} {(4 i)}^{3} + 5 \cdot 4 {(4 i)}^{4} + {(4 i)}^{5}$

Step 2.1.6

Multiply $10$ by $64$ .

$1024 + 5120 i + 640 {(4 i)}^{2} + 10 \cdot 4^{2} {(4 i)}^{3} + 5 \cdot 4 {(4 i)}^{4} + {(4 i)}^{5}$

Step 2.1.7

Apply the product rule to $4 i$ .

$1024 + 5120 i + 640 (4^{2} i^{2}) + 10 \cdot 4^{2} {(4 i)}^{3} + 5 \cdot 4 {(4 i)}^{4} + {(4 i)}^{5}$

Step 2.1.8

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

$1024 + 5120 i + 640 (16 i^{2}) + 10 \cdot 4^{2} {(4 i)}^{3} + 5 \cdot 4 {(4 i)}^{4} + {(4 i)}^{5}$

Step 2.1.9

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

$1024 + 5120 i + 640 (16 \cdot - 1) + 10 \cdot 4^{2} {(4 i)}^{3} + 5 \cdot 4 {(4 i)}^{4} + {(4 i)}^{5}$

Step 2.1.10

Multiply $640 (16 \cdot - 1)$ .

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Step 2.1.10.1

Multiply $16$ by $- 1$ .

$1024 + 5120 i + 640 \cdot - 16 + 10 \cdot 4^{2} {(4 i)}^{3} + 5 \cdot 4 {(4 i)}^{4} + {(4 i)}^{5}$

Step 2.1.10.2

Multiply $640$ by $- 16$ .

$1024 + 5120 i - 10240 + 10 \cdot 4^{2} {(4 i)}^{3} + 5 \cdot 4 {(4 i)}^{4} + {(4 i)}^{5}$

Step 2.1.11

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

$1024 + 5120 i - 10240 + 10 \cdot 16 {(4 i)}^{3} + 5 \cdot 4 {(4 i)}^{4} + {(4 i)}^{5}$

Step 2.1.12

Multiply $10$ by $16$ .

$1024 + 5120 i - 10240 + 160 {(4 i)}^{3} + 5 \cdot 4 {(4 i)}^{4} + {(4 i)}^{5}$

Step 2.1.13

Apply the product rule to $4 i$ .

$1024 + 5120 i - 10240 + 160 (4^{3} i^{3}) + 5 \cdot 4 {(4 i)}^{4} + {(4 i)}^{5}$

Step 2.1.14

Raise $4$ to the power of $3$ .

$1024 + 5120 i - 10240 + 160 (64 i^{3}) + 5 \cdot 4 {(4 i)}^{4} + {(4 i)}^{5}$

Step 2.1.15

Factor out $i^{2}$ .

$1024 + 5120 i - 10240 + 160 (64 (i^{2} \cdot i)) + 5 \cdot 4 {(4 i)}^{4} + {(4 i)}^{5}$

Step 2.1.16

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

$1024 + 5120 i - 10240 + 160 (64 (- 1 \cdot i)) + 5 \cdot 4 {(4 i)}^{4} + {(4 i)}^{5}$

Step 2.1.17

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

$1024 + 5120 i - 10240 + 160 (64 (- i)) + 5 \cdot 4 {(4 i)}^{4} + {(4 i)}^{5}$

Step 2.1.18

Multiply $- 1$ by $64$ .

$1024 + 5120 i - 10240 + 160 (- 64 i) + 5 \cdot 4 {(4 i)}^{4} + {(4 i)}^{5}$

Step 2.1.19

Multiply $- 64$ by $160$ .

$1024 + 5120 i - 10240 - 10240 i + 5 \cdot 4 {(4 i)}^{4} + {(4 i)}^{5}$

Step 2.1.20

Multiply $5$ by $4$ .

$1024 + 5120 i - 10240 - 10240 i + 20 {(4 i)}^{4} + {(4 i)}^{5}$

Step 2.1.21

Apply the product rule to $4 i$ .

$1024 + 5120 i - 10240 - 10240 i + 20 (4^{4} i^{4}) + {(4 i)}^{5}$

Step 2.1.22

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

$1024 + 5120 i - 10240 - 10240 i + 20 (256 i^{4}) + {(4 i)}^{5}$

Step 2.1.23

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

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Step 2.1.23.1

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

$1024 + 5120 i - 10240 - 10240 i + 20 (256 {(i^{2})}^{2}) + {(4 i)}^{5}$

Step 2.1.23.2

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

$1024 + 5120 i - 10240 - 10240 i + 20 (256 {(- 1)}^{2}) + {(4 i)}^{5}$

Step 2.1.23.3

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

$1024 + 5120 i - 10240 - 10240 i + 20 (256 \cdot 1) + {(4 i)}^{5}$

Step 2.1.24

Multiply $20 (256 \cdot 1)$ .

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Step 2.1.24.1

Multiply $256$ by $1$ .

$1024 + 5120 i - 10240 - 10240 i + 20 \cdot 256 + {(4 i)}^{5}$

Step 2.1.24.2

Multiply $20$ by $256$ .

$1024 + 5120 i - 10240 - 10240 i + 5120 + {(4 i)}^{5}$

Step 2.1.25

Apply the product rule to $4 i$ .

$1024 + 5120 i - 10240 - 10240 i + 5120 + 4^{5} i^{5}$

Step 2.1.26

Raise $4$ to the power of $5$ .

$1024 + 5120 i - 10240 - 10240 i + 5120 + 1024 i^{5}$

Step 2.1.27

Factor out $i^{4}$ .

$1024 + 5120 i - 10240 - 10240 i + 5120 + 1024 (i^{4} i)$

Step 2.1.28

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

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Step 2.1.28.1

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

$1024 + 5120 i - 10240 - 10240 i + 5120 + 1024 ({(i^{2})}^{2} i)$

Step 2.1.28.2

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

$1024 + 5120 i - 10240 - 10240 i + 5120 + 1024 ({(- 1)}^{2} i)$

Step 2.1.28.3

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

$1024 + 5120 i - 10240 - 10240 i + 5120 + 1024 (1 i)$

Step 2.1.29

Multiply $i$ by $1$ .

$1024 + 5120 i - 10240 - 10240 i + 5120 + 1024 i$

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Subtract $10240$ from $1024$ .

$- 9216 + 5120 i - 10240 i + 5120 + 1024 i$

Step 2.2.2

Add $- 9216$ and $5120$ .

$- 4096 + 5120 i - 10240 i + 1024 i$

Step 2.2.3

Subtract $10240 i$ from $5120 i$ .

$- 4096 - 5120 i + 1024 i$

Step 2.2.4

Add $- 5120 i$ and $1024 i$ .

$- 4096 - 4096 i$

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 $a = - 4096$ and $b = - 4096$ .

$| z | = \sqrt{{(- 4096)}^{2} + {(- 4096)}^{2}}$

Step 6

Find $| z |$ .

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Step 6.1

Raise $- 4096$ to the power of $2$ .

$| z | = \sqrt{16777216 + {(- 4096)}^{2}}$

Step 6.2

Raise $- 4096$ to the power of $2$ .

$| z | = \sqrt{16777216 + 16777216}$

Step 6.3

Add $16777216$ and $16777216$ .

$| z | = \sqrt{33554432}$

Step 6.4

Rewrite $33554432$ as $4096^{2} \cdot 2$ .

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Step 6.4.1

Factor $16777216$ out of $33554432$ .

$| z | = \sqrt{16777216 (2)}$

Step 6.4.2

Rewrite $16777216$ as $4096^{2}$ .

$| z | = \sqrt{4096^{2} \cdot 2}$

Step 6.5

Pull terms out from under the radical.

$| z | = 4096 \sqrt{2}$

Step 7

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

$θ = \arctan (\frac{- 4096}{- 4096})$

Step 8

Since inverse tangent of $\frac{- 4096}{- 4096}$ produces an angle in the third quadrant, the value of the angle is $\frac{5 π}{4}$ .

$θ = \frac{5 π}{4}$

Step 9

Substitute the values of $θ = \frac{5 π}{4}$ and $| z | = 4096 \sqrt{2}$ .

$4096 \sqrt{2} (\cos (\frac{5 π}{4}) + i \sin (\frac{5 π}{4}))$