Linear Algebra Examples

$\sqrt{5} + i \sqrt{5}$

Step 1

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 2

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 3

Substitute the actual values of $a = \sqrt{5}$ and $b = \sqrt{5}$ .

$| z | = \sqrt{{(\sqrt{5})}^{2} + {(\sqrt{5})}^{2}}$

Step 4

Find $| z |$ .

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Step 4.1

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 4.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$| z | = \sqrt{{(5^{\frac{1}{2}})}^{2} + {(\sqrt{5})}^{2}}$

Step 4.1.2

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

$| z | = \sqrt{5^{\frac{1}{2} \cdot 2} + {(\sqrt{5})}^{2}}$

Step 4.1.3

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

$| z | = \sqrt{5^{\frac{2}{2}} + {(\sqrt{5})}^{2}}$

Step 4.1.4

Cancel the common factor of $2$ .

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Step 4.1.4.1

Cancel the common factor.

$| z | = \sqrt{5^{\frac{2}{2}} + {(\sqrt{5})}^{2}}$

Step 4.1.4.2

Rewrite the expression.

$| z | = \sqrt{5 + {(\sqrt{5})}^{2}}$

Step 4.1.5

Evaluate the exponent.

$| z | = \sqrt{5 + {(\sqrt{5})}^{2}}$

Step 4.2

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 4.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$| z | = \sqrt{5 + {(5^{\frac{1}{2}})}^{2}}$

Step 4.2.2

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

$| z | = \sqrt{5 + 5^{\frac{1}{2} \cdot 2}}$

Step 4.2.3

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

$| z | = \sqrt{5 + 5^{\frac{2}{2}}}$

Step 4.2.4

Cancel the common factor of $2$ .

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Step 4.2.4.1

Cancel the common factor.

$| z | = \sqrt{5 + 5^{\frac{2}{2}}}$

Step 4.2.4.2

Rewrite the expression.

$| z | = \sqrt{5 + 5}$

Step 4.2.5

Evaluate the exponent.

$| z | = \sqrt{5 + 5}$

Step 4.3

Add $5$ and $5$ .

$| z | = \sqrt{10}$

Step 5

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

$θ = \arctan (\frac{\sqrt{5}}{\sqrt{5}})$

Step 6

Since inverse tangent of $\frac{\sqrt{5}}{\sqrt{5}}$ produces an angle in the first quadrant, the value of the angle is $\frac{π}{4}$ .

$θ = \frac{π}{4}$

Step 7

Substitute the values of $θ = \frac{π}{4}$ and $| z | = \sqrt{10}$ .

$\sqrt{10} (\cos (\frac{π}{4}) + i \sin (\frac{π}{4}))$