Linear Algebra Examples

$(8 - 7 i) (6 + 7 i)$

Step 1

Expand $(8 - 7 i) (6 + 7 i)$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

$8 (6 + 7 i) - 7 i (6 + 7 i)$

Step 1.2

Apply the distributive property.

$8 \cdot 6 + 8 (7 i) - 7 i (6 + 7 i)$

Step 1.3

Apply the distributive property.

$8 \cdot 6 + 8 (7 i) - 7 i \cdot 6 - 7 i (7 i)$

Step 2

Simplify and combine like terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Multiply $8$ by $6$ .

$48 + 8 (7 i) - 7 i \cdot 6 - 7 i (7 i)$

Step 2.1.2

Multiply $7$ by $8$ .

$48 + 56 i - 7 i \cdot 6 - 7 i (7 i)$

Step 2.1.3

Multiply $6$ by $- 7$ .

$48 + 56 i - 42 i - 7 i (7 i)$

Step 2.1.4

Multiply $- 7 i (7 i)$ .

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Step 2.1.4.1

Multiply $7$ by $- 7$ .

$48 + 56 i - 42 i - 49 i i$

Step 2.1.4.2

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

$48 + 56 i - 42 i - 49 (i^{1} i)$

Step 2.1.4.3

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

$48 + 56 i - 42 i - 49 (i^{1} i^{1})$

Step 2.1.4.4

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

$48 + 56 i - 42 i - 49 i^{1 + 1}$

Step 2.1.4.5

Add $1$ and $1$ .

$48 + 56 i - 42 i - 49 i^{2}$

Step 2.1.5

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

$48 + 56 i - 42 i - 49 \cdot - 1$

Step 2.1.6

Multiply $- 49$ by $- 1$ .

$48 + 56 i - 42 i + 49$

Step 2.2

Add $48$ and $49$ .

$97 + 56 i - 42 i$

Step 2.3

Subtract $42 i$ from $56 i$ .

$97 + 14 i$

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 $a = 97$ and $b = 14$ .

$| z | = \sqrt{14^{2} + 97^{2}}$

Step 6

Find $| z |$ .

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Step 6.1

Raise $14$ to the power of $2$ .

$| z | = \sqrt{196 + 97^{2}}$

Step 6.2

Raise $97$ to the power of $2$ .

$| z | = \sqrt{196 + 9409}$

Step 6.3

Add $196$ and $9409$ .

$| z | = \sqrt{9605}$

Step 7

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

$θ = \arctan (\frac{14}{97})$

Step 8

Since inverse tangent of $\frac{14}{97}$ produces an angle in the first quadrant, the value of the angle is $0.14334005$ .

$θ = 0.14334005$

Step 9

Substitute the values of $θ = 0.14334005$ and $| z | = \sqrt{9605}$ .

$\sqrt{9605} (\cos (0.14334005) + i \sin (0.14334005))$