Calculus Examples

$(- 5 + 27 i) - (- 12 i + 8) + {(- 2 i)}^{3}$

Step 1

Simplify each term.

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Step 1.1

Apply the distributive property.

$- 5 + 27 i - (- 12 i) - 1 \cdot 8 + {(- 2 i)}^{3}$

Step 1.2

Multiply $- 12$ by $- 1$ .

$- 5 + 27 i + 12 i - 1 \cdot 8 + {(- 2 i)}^{3}$

Step 1.3

Multiply $- 1$ by $8$ .

$- 5 + 27 i + 12 i - 8 + {(- 2 i)}^{3}$

Step 1.4

Apply the product rule to $- 2 i$ .

$- 5 + 27 i + 12 i - 8 + {(- 2)}^{3} i^{3}$

Step 1.5

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

$- 5 + 27 i + 12 i - 8 - 8 i^{3}$

Step 1.6

Factor out $i^{2}$ .

$- 5 + 27 i + 12 i - 8 - 8 (i^{2} \cdot i)$

Step 1.7

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

$- 5 + 27 i + 12 i - 8 - 8 (- 1 \cdot i)$

Step 1.8

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

$- 5 + 27 i + 12 i - 8 - 8 (- i)$

Step 1.9

Multiply $- 1$ by $- 8$ .

$- 5 + 27 i + 12 i - 8 + 8 i$

Step 2

Simplify by adding terms.

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Step 2.1

Subtract $8$ from $- 5$ .

$- 13 + 27 i + 12 i + 8 i$

Step 2.2

Add $27 i$ and $12 i$ .

$- 13 + 39 i + 8 i$

Step 2.3

Add $39 i$ and $8 i$ .

$- 13 + 47 i$

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 $a = - 13$ and $b = 47$ .

$| z | = \sqrt{47^{2} + {(- 13)}^{2}}$

Step 6

Find $| z |$ .

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Step 6.1

Raise $47$ to the power of $2$ .

$| z | = \sqrt{2209 + {(- 13)}^{2}}$

Step 6.2

Raise $- 13$ to the power of $2$ .

$| z | = \sqrt{2209 + 169}$

Step 6.3

Add $2209$ and $169$ .

$| z | = \sqrt{2378}$

Step 7

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

$θ = \arctan (\frac{47}{- 13})$

Step 8

Since inverse tangent of $\frac{47}{- 13}$ produces an angle in the second quadrant, the value of the angle is $1.84064548$ .

$θ = 1.84064548$

Step 9

Substitute the values of $θ = 1.84064548$ and $| z | = \sqrt{2378}$ .

$\sqrt{2378} (\cos (1.84064548) + i \sin (1.84064548))$