Algebra Examples

$\sqrt[3]{2} (\cos (120) + i \sin (120))$

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$\sqrt[3]{2} (- \cos (60) + i \sin (120))$

Step 1.2

The exact value of $\cos (60)$ is $\frac{1}{2}$ .

$\sqrt[3]{2} (- \frac{1}{2} + i \sin (120))$

Step 1.3

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$\sqrt[3]{2} (- \frac{1}{2} + i \sin (60))$

Step 1.4

The exact value of $\sin (60)$ is $\frac{\sqrt{3}}{2}$ .

$\sqrt[3]{2} (- \frac{1}{2} + i \frac{\sqrt{3}}{2})$

Step 1.5

Combine $i$ and $\frac{\sqrt{3}}{2}$ .

$\sqrt[3]{2} (- \frac{1}{2} + \frac{i \sqrt{3}}{2})$

Step 2

Simplify terms.

Tap for more steps...

Step 2.1

Apply the distributive property.

$\sqrt[3]{2} (- \frac{1}{2}) + \sqrt[3]{2} \frac{i \sqrt{3}}{2}$

Step 2.2

Combine $\sqrt[3]{2}$ and $\frac{1}{2}$ .

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

Step 3

Multiply $\sqrt[3]{2} \frac{i \sqrt{3}}{2}$ .

Tap for more steps...

Step 3.1

Combine $\sqrt[3]{2}$ and $\frac{i \sqrt{3}}{2}$ .

$- \frac{\sqrt[3]{2}}{2} + \frac{\sqrt[3]{2} (i \sqrt{3})}{2}$

Step 3.2

Rewrite the expression using the least common index of $6$ .

Tap for more steps...

Step 3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$- \frac{\sqrt[3]{2}}{2} + \frac{3^{\frac{1}{2}} \sqrt[3]{2} i}{2}$

Step 3.2.2

Rewrite $3^{\frac{1}{2}}$ as $3^{\frac{3}{6}}$ .

$- \frac{\sqrt[3]{2}}{2} + \frac{3^{\frac{3}{6}} \sqrt[3]{2} i}{2}$

Step 3.2.3

Rewrite $3^{\frac{3}{6}}$ as $\sqrt[6]{3^{3}}$ .

$- \frac{\sqrt[3]{2}}{2} + \frac{\sqrt[6]{3^{3}} \sqrt[3]{2} i}{2}$

Step 3.2.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{2}$ as $2^{\frac{1}{3}}$ .

$- \frac{\sqrt[3]{2}}{2} + \frac{\sqrt[6]{3^{3}} \cdot 2^{\frac{1}{3}} i}{2}$

Step 3.2.5

Rewrite $2^{\frac{1}{3}}$ as $2^{\frac{2}{6}}$ .

$- \frac{\sqrt[3]{2}}{2} + \frac{\sqrt[6]{3^{3}} \cdot 2^{\frac{2}{6}} i}{2}$

Step 3.2.6

Rewrite $2^{\frac{2}{6}}$ as $\sqrt[6]{2^{2}}$ .

$- \frac{\sqrt[3]{2}}{2} + \frac{\sqrt[6]{3^{3}} \sqrt[6]{2^{2}} i}{2}$

Step 3.3

Combine using the product rule for radicals.

$- \frac{\sqrt[3]{2}}{2} + \frac{\sqrt[6]{3^{3} \cdot 2^{2}} i}{2}$

Step 3.4

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

$- \frac{\sqrt[3]{2}}{2} + \frac{\sqrt[6]{27 \cdot 2^{2}} i}{2}$

Step 3.5

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

$- \frac{\sqrt[3]{2}}{2} + \frac{\sqrt[6]{27 \cdot 4} i}{2}$

Step 3.6

Multiply $27$ by $4$ .

$- \frac{\sqrt[3]{2}}{2} + \frac{\sqrt[6]{108} i}{2}$