Precalculus Examples

$4 i \cdot (5 i)$

Step 1

Multiply $4 i (5 i)$ .

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Step 1.1

Multiply $5$ by $4$ .

$20 i i$

Step 1.2

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

$20 (i^{1} i)$

Step 1.3

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

$20 (i^{1} i^{1})$

Step 1.4

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

$20 i^{1 + 1}$

Step 1.5

Add $1$ and $1$ .

$20 i^{2}$

Step 2

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

$20 \cdot - 1$

Step 3

Multiply $20$ by $- 1$ .

$- 20$

Step 4

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 5

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 6

Substitute the actual values of $a = - 20$ and $b = 0$ .

$| z | = \sqrt{0^{2} + {(- 20)}^{2}}$

Step 7

Find $| z |$ .

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Step 7.1

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

$| z | = \sqrt{0 + {(- 20)}^{2}}$

Step 7.2

Raise $- 20$ to the power of $2$ .

$| z | = \sqrt{0 + 400}$

Step 7.3

Add $0$ and $400$ .

$| z | = \sqrt{400}$

Step 7.4

Rewrite $400$ as $20^{2}$ .

$| z | = \sqrt{20^{2}}$

Step 7.5

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

$| z | = 20$

Step 8

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

$θ = \arctan (\frac{0}{- 20})$

Step 9

Since inverse tangent of $\frac{0}{- 20}$ produces an angle in the second quadrant, the value of the angle is $π$ .

$θ = π$

Step 10

Substitute the values of $θ = π$ and $| z | = 20$ .

$20 (\cos (π) + i \sin (π))$