Linear Algebra Examples

$i^{8}$

Step 1

Rewrite $i^{8}$ as ${(i^{4})}^{2}$ .

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

${({(- 1)}^{2})}^{2}$

Step 2.3

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

$1^{2}$

Step 3

One to any power is one.

$1$

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 $a = 1$ and $b = 0$ .

$| z | = \sqrt{0^{2} + 1^{2}}$

Step 7

Find $| z |$ .

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Step 7.1

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

$| z | = \sqrt{0 + 1^{2}}$

Step 7.2

One to any power is one.

$| z | = \sqrt{0 + 1}$

Step 7.3

Add $0$ and $1$ .

$| z | = \sqrt{1}$

Step 7.4

Any root of $1$ is $1$ .

$| z | = 1$

Step 8

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

$θ = \arctan (\frac{0}{1})$

Step 9

Since inverse tangent of $\frac{0}{1}$ produces an angle in the first quadrant, the value of the angle is $0$ .

$θ = 0$

Step 10

Substitute the values of $θ = 0$ and $| z | = 1$ .

$\cos (0) + i \sin (0)$