Algebra Examples

$- \frac{\sqrt{2}}{3} + i$

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 $a = - \frac{\sqrt{2}}{3}$ and $b = 1$ .

$| z | = \sqrt{1^{2} + {(- \frac{\sqrt{2}}{3})}^{2}}$

Step 4

Find $| z |$ .

Tap for more steps...

Step 4.1

One to any power is one.

$| z | = \sqrt{1 + {(- \frac{\sqrt{2}}{3})}^{2}}$

Step 4.2

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

Tap for more steps...

Step 4.2.1

Apply the product rule to $- \frac{\sqrt{2}}{3}$ .

$| z | = \sqrt{1 + {(- 1)}^{2} {(\frac{\sqrt{2}}{3})}^{2}}$

Step 4.2.2

Apply the product rule to $\frac{\sqrt{2}}{3}$ .

$| z | = \sqrt{1 + {(- 1)}^{2} (\frac{{\sqrt{2}}^{2}}{3^{2}})}$

Step 4.3

Simplify the expression.

Tap for more steps...

Step 4.3.1

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

$| z | = \sqrt{1 + 1 (\frac{{\sqrt{2}}^{2}}{3^{2}})}$

Step 4.3.2

Multiply $\frac{{\sqrt{2}}^{2}}{3^{2}}$ by $1$ .

$| z | = \sqrt{1 + \frac{{\sqrt{2}}^{2}}{3^{2}}}$

Step 4.4

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

Tap for more steps...

Step 4.4.1

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

$| z | = \sqrt{1 + \frac{{(2^{\frac{1}{2}})}^{2}}{3^{2}}}$

Step 4.4.2

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

$| z | = \sqrt{1 + \frac{2^{\frac{1}{2} \cdot 2}}{3^{2}}}$

Step 4.4.3

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

$| z | = \sqrt{1 + \frac{2^{\frac{2}{2}}}{3^{2}}}$

Step 4.4.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.4.4.1

Cancel the common factor.

$| z | = \sqrt{1 + \frac{2^{\frac{2}{2}}}{3^{2}}}$

Step 4.4.4.2

Rewrite the expression.

$| z | = \sqrt{1 + \frac{2}{3^{2}}}$

Step 4.4.5

Evaluate the exponent.

$| z | = \sqrt{1 + \frac{2}{3^{2}}}$

Step 4.5

Simplify the expression.

Tap for more steps...

Step 4.5.1

Raise $3$ to the power of $2$ .

$| z | = \sqrt{1 + \frac{2}{9}}$

Step 4.5.2

Write $1$ as a fraction with a common denominator.

$| z | = \sqrt{\frac{9}{9} + \frac{2}{9}}$

Step 4.5.3

Combine the numerators over the common denominator.

$| z | = \sqrt{\frac{9 + 2}{9}}$

Step 4.5.4

Add $9$ and $2$ .

$| z | = \sqrt{\frac{11}{9}}$

Step 4.6

Rewrite $\sqrt{\frac{11}{9}}$ as $\frac{\sqrt{11}}{\sqrt{9}}$ .

$| z | = \frac{\sqrt{11}}{\sqrt{9}}$

Step 4.7

Simplify the denominator.

Tap for more steps...

Step 4.7.1

Rewrite $9$ as $3^{2}$ .

$| z | = \frac{\sqrt{11}}{\sqrt{3^{2}}}$

Step 4.7.2

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

$| z | = \frac{\sqrt{11}}{3}$

Step 5

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

$θ = \arctan (\frac{1}{- \frac{\sqrt{2}}{3}})$

Step 6

Since inverse tangent of $\frac{1}{- \frac{\sqrt{2}}{3}}$ produces an angle in the second quadrant, the value of the angle is $2.01130698$ .

$θ = 2.01130698$

Step 7

Substitute the values of $θ = 2.01130698$ and $| z | = \frac{\sqrt{11}}{3}$ .

$\frac{\sqrt{11}}{3 (\cos (2.01130698) + i \sin (2.01130698))}$