Linear Algebra Examples

$8 i (7 - 5 i)$

Step 1

Apply the distributive property.

$8 i \cdot 7 + 8 i (- 5 i)$

Step 2

Multiply $7$ by $8$ .

$56 i + 8 i (- 5 i)$

Step 3

Multiply $8 i (- 5 i)$ .

Tap for more steps...

Step 3.1

Multiply $- 5$ by $8$ .

$56 i - 40 i i$

Step 3.2

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

$56 i - 40 (i^{1} i)$

Step 3.3

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

$56 i - 40 (i^{1} i^{1})$

Step 3.4

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

$56 i - 40 i^{1 + 1}$

Step 3.5

Add $1$ and $1$ .

$56 i - 40 i^{2}$

Step 4

Simplify each term.

Tap for more steps...

Step 4.1

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

$56 i - 40 \cdot - 1$

Step 4.2

Multiply $- 40$ by $- 1$ .

$56 i + 40$

Step 5

Reorder $56 i$ and $40$ .

$40 + 56 i$

Step 6

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 7

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 8

Substitute the actual values of $a = 40$ and $b = 56$ .

$| z | = \sqrt{56^{2} + 40^{2}}$

Step 9

Find $| z |$ .

Tap for more steps...

Step 9.1

Raise $56$ to the power of $2$ .

$| z | = \sqrt{3136 + 40^{2}}$

Step 9.2

Raise $40$ to the power of $2$ .

$| z | = \sqrt{3136 + 1600}$

Step 9.3

Add $3136$ and $1600$ .

$| z | = \sqrt{4736}$

Step 9.4

Rewrite $4736$ as $8^{2} \cdot 74$ .

Tap for more steps...

Step 9.4.1

Factor $64$ out of $4736$ .

$| z | = \sqrt{64 (74)}$

Step 9.4.2

Rewrite $64$ as $8^{2}$ .

$| z | = \sqrt{8^{2} \cdot 74}$

Step 9.5

Pull terms out from under the radical.

$| z | = 8 \sqrt{74}$

Step 10

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

$θ = \arctan (\frac{56}{40})$

Step 11

Since inverse tangent of $\frac{56}{40}$ produces an angle in the first quadrant, the value of the angle is $0.95054684$ .

$θ = 0.95054684$

Step 12

Substitute the values of $θ = 0.95054684$ and $| z | = 8 \sqrt{74}$ .

$8 \sqrt{74} (\cos (0.95054684) + i \sin (0.95054684))$