Linear Algebra Examples

$(\frac{6}{6 + i}) (\frac{6 - i}{6 - i})$

Step 1

Cancel the common factor of $6 - i$ .

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Step 1.1

Cancel the common factor.

$\frac{6}{6 + i} \cdot \frac{6 - i}{6 - i}$

Step 1.2

Rewrite the expression.

$\frac{6}{6 + i} \cdot 1$

Step 2

Multiply $\frac{6}{6 + i}$ by $1$ .

$\frac{6}{6 + i}$

Step 3

Multiply the numerator and denominator of $\frac{6}{6 + 1 i}$ by the conjugate of $6 + 1 i$ to make the denominator real.

$\frac{6}{6 + 1 i} \cdot \frac{6 - i}{6 - i}$

Step 4

Multiply.

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Step 4.1

Combine.

$\frac{6 (6 - i)}{(6 + 1 i) (6 - i)}$

Step 4.2

Simplify the numerator.

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Step 4.2.1

Apply the distributive property.

$\frac{6 \cdot 6 + 6 (- i)}{(6 + 1 i) (6 - i)}$

Step 4.2.2

Multiply $6$ by $6$ .

$\frac{36 + 6 (- i)}{(6 + 1 i) (6 - i)}$

Step 4.2.3

Multiply $- 1$ by $6$ .

$\frac{36 - 6 i}{(6 + 1 i) (6 - i)}$

Step 4.3

Simplify the denominator.

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Step 4.3.1

Expand $(6 + 1 i) (6 - i)$ using the FOIL Method.

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Step 4.3.1.1

Apply the distributive property.

$\frac{36 - 6 i}{6 (6 - i) + 1 i (6 - i)}$

Step 4.3.1.2

Apply the distributive property.

$\frac{36 - 6 i}{6 \cdot 6 + 6 (- i) + 1 i (6 - i)}$

Step 4.3.1.3

Apply the distributive property.

$\frac{36 - 6 i}{6 \cdot 6 + 6 (- i) + 1 i \cdot 6 + 1 i (- i)}$

Step 4.3.2

Simplify.

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Step 4.3.2.1

Multiply $6$ by $6$ .

$\frac{36 - 6 i}{36 + 6 (- i) + 1 i \cdot 6 + 1 i (- i)}$

Step 4.3.2.2

Multiply $- 1$ by $6$ .

$\frac{36 - 6 i}{36 - 6 i + 1 i \cdot 6 + 1 i (- i)}$

Step 4.3.2.3

Multiply $6$ by $1$ .

$\frac{36 - 6 i}{36 - 6 i + 6 i + 1 i (- i)}$

Step 4.3.2.4

Multiply $- 1$ by $1$ .

$\frac{36 - 6 i}{36 - 6 i + 6 i - i i}$

Step 4.3.2.5

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

$\frac{36 - 6 i}{36 - 6 i + 6 i - (i^{1} i)}$

Step 4.3.2.6

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

$\frac{36 - 6 i}{36 - 6 i + 6 i - (i^{1} i^{1})}$

Step 4.3.2.7

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

$\frac{36 - 6 i}{36 - 6 i + 6 i - i^{1 + 1}}$

Step 4.3.2.8

Add $1$ and $1$ .

$\frac{36 - 6 i}{36 - 6 i + 6 i - i^{2}}$

Step 4.3.2.9

Add $- 6 i$ and $6 i$ .

$\frac{36 - 6 i}{36 + 0 - i^{2}}$

Step 4.3.2.10

Add $36$ and $0$ .

$\frac{36 - 6 i}{36 - i^{2}}$

Step 4.3.3

Simplify each term.

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Step 4.3.3.1

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

$\frac{36 - 6 i}{36 - - 1}$

Step 4.3.3.2

Multiply $- 1$ by $- 1$ .

$\frac{36 - 6 i}{36 + 1}$

Step 4.3.4

Add $36$ and $1$ .

$\frac{36 - 6 i}{37}$

Step 5

Split the fraction $\frac{36 - 6 i}{37}$ into two fractions.

$\frac{36}{37} + \frac{- 6 i}{37}$

Step 6

Move the negative in front of the fraction.

$\frac{36}{37} - \frac{6 i}{37}$

Step 7

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 8

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 9

Substitute the actual values of $a = \frac{36}{37}$ and $b = - \frac{6}{37}$ .

$| z | = \sqrt{{(- \frac{6}{37})}^{2} + {(\frac{36}{37})}^{2}}$

Step 10

Find $| z |$ .

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Step 10.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 10.1.1

Apply the product rule to $- \frac{6}{37}$ .

$| z | = \sqrt{{(- 1)}^{2} {(\frac{6}{37})}^{2} + {(\frac{36}{37})}^{2}}$

Step 10.1.2

Apply the product rule to $\frac{6}{37}$ .

$| z | = \sqrt{{(- 1)}^{2} (\frac{6^{2}}{37^{2}}) + {(\frac{36}{37})}^{2}}$

Step 10.2

Simplify the expression.

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Step 10.2.1

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

$| z | = \sqrt{1 (\frac{6^{2}}{37^{2}}) + {(\frac{36}{37})}^{2}}$

Step 10.2.2

Multiply $\frac{6^{2}}{37^{2}}$ by $1$ .

$| z | = \sqrt{\frac{6^{2}}{37^{2}} + {(\frac{36}{37})}^{2}}$

Step 10.2.3

Raise $6$ to the power of $2$ .

$| z | = \sqrt{\frac{36}{37^{2}} + {(\frac{36}{37})}^{2}}$

Step 10.2.4

Raise $37$ to the power of $2$ .

$| z | = \sqrt{\frac{36}{1369} + {(\frac{36}{37})}^{2}}$

Step 10.2.5

Apply the product rule to $\frac{36}{37}$ .

$| z | = \sqrt{\frac{36}{1369} + \frac{36^{2}}{37^{2}}}$

Step 10.2.6

Raise $36$ to the power of $2$ .

$| z | = \sqrt{\frac{36}{1369} + \frac{1296}{37^{2}}}$

Step 10.2.7

Raise $37$ to the power of $2$ .

$| z | = \sqrt{\frac{36}{1369} + \frac{1296}{1369}}$

Step 10.2.8

Combine the numerators over the common denominator.

$| z | = \sqrt{\frac{36 + 1296}{1369}}$

Step 10.2.9

Add $36$ and $1296$ .

$| z | = \sqrt{\frac{1332}{1369}}$

Step 10.3

Cancel the common factor of $1332$ and $1369$ .

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Step 10.3.1

Factor $37$ out of $1332$ .

$| z | = \sqrt{\frac{37 (36)}{1369}}$

Step 10.3.2

Cancel the common factors.

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Step 10.3.2.1

Factor $37$ out of $1369$ .

$| z | = \sqrt{\frac{37 \cdot 36}{37 \cdot 37}}$

Step 10.3.2.2

Cancel the common factor.

$| z | = \sqrt{\frac{37 \cdot 36}{37 \cdot 37}}$

Step 10.3.2.3

Rewrite the expression.

$| z | = \sqrt{\frac{36}{37}}$

Step 10.4

Rewrite $\sqrt{\frac{36}{37}}$ as $\frac{\sqrt{36}}{\sqrt{37}}$ .

$| z | = \frac{\sqrt{36}}{\sqrt{37}}$

Step 10.5

Simplify the numerator.

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Step 10.5.1

Rewrite $36$ as $6^{2}$ .

$| z | = \frac{\sqrt{6^{2}}}{\sqrt{37}}$

Step 10.5.2

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

$| z | = \frac{6}{\sqrt{37}}$

Step 10.6

Multiply $\frac{6}{\sqrt{37}}$ by $\frac{\sqrt{37}}{\sqrt{37}}$ .

$| z | = \frac{6}{\sqrt{37}} \cdot \frac{\sqrt{37}}{\sqrt{37}}$

Step 10.7

Combine and simplify the denominator.

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Step 10.7.1

Multiply $\frac{6}{\sqrt{37}}$ by $\frac{\sqrt{37}}{\sqrt{37}}$ .

$| z | = \frac{6 \sqrt{37}}{\sqrt{37} \sqrt{37}}$

Step 10.7.2

Raise $\sqrt{37}$ to the power of $1$ .

$| z | = \frac{6 \sqrt{37}}{\sqrt{37} \sqrt{37}}$

Step 10.7.3

Raise $\sqrt{37}$ to the power of $1$ .

$| z | = \frac{6 \sqrt{37}}{\sqrt{37} \sqrt{37}}$

Step 10.7.4

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

$| z | = \frac{6 \sqrt{37}}{{\sqrt{37}}^{1 + 1}}$

Step 10.7.5

Add $1$ and $1$ .

$| z | = \frac{6 \sqrt{37}}{{\sqrt{37}}^{2}}$

Step 10.7.6

Rewrite ${\sqrt{37}}^{2}$ as $37$ .

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Step 10.7.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{37}$ as $37^{\frac{1}{2}}$ .

$| z | = \frac{6 \sqrt{37}}{{(37^{\frac{1}{2}})}^{2}}$

Step 10.7.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$| z | = \frac{6 \sqrt{37}}{37^{\frac{1}{2} \cdot 2}}$

Step 10.7.6.3

Combine $\frac{1}{2}$ and $2$ .

$| z | = \frac{6 \sqrt{37}}{37^{\frac{2}{2}}}$

Step 10.7.6.4

Cancel the common factor of $2$ .

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Step 10.7.6.4.1

Cancel the common factor.

$| z | = \frac{6 \sqrt{37}}{37^{\frac{2}{2}}}$

Step 10.7.6.4.2

Rewrite the expression.

$| z | = \frac{6 \sqrt{37}}{37}$

Step 10.7.6.5

Evaluate the exponent.

$| z | = \frac{6 \sqrt{37}}{37}$

Step 11

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

$θ = \arctan (\frac{- \frac{6}{37}}{\frac{36}{37}})$

Step 12

Since inverse tangent of $\frac{- \frac{6}{37}}{\frac{36}{37}}$ produces an angle in the fourth quadrant, the value of the angle is $- 0.16514867$ .

$θ = - 0.16514867$

Step 13

Substitute the values of $θ = - 0.16514867$ and $| z | = \frac{6 \sqrt{37}}{37}$ .

$\frac{6 \sqrt{37}}{37 (\cos (- 0.16514867) + i \sin (- 0.16514867))}$