Precalculus Examples

$\sqrt[3]{125} (\cos (\frac{150 + 360 \cdot 0}{3}) + i \sin (\frac{150 + 360 \cdot 0}{3}))$

Step 1

Rewrite $125$ as $5^{3}$ .

$\sqrt[3]{5^{3}} (\cos (\frac{150 + 360 \cdot 0}{3}) + i \sin (\frac{150 + 360 \cdot 0}{3}))$

Step 2

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

$5 (\cos (\frac{150 + 360 \cdot 0}{3}) + i \sin (\frac{150 + 360 \cdot 0}{3}))$

Step 3

Simplify each term.

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Step 3.1

Cancel the common factor of $150 + 360 \cdot 0$ and $3$ .

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Step 3.1.1

Factor $3$ out of $150$ .

$5 (\cos (\frac{3 \cdot 50 + 360 \cdot 0}{3}) + i \sin (\frac{150 + 360 \cdot 0}{3}))$

Step 3.1.2

Factor $3$ out of $360 \cdot 0$ .

$5 (\cos (\frac{3 \cdot 50 + 3 (120 \cdot 0)}{3}) + i \sin (\frac{150 + 360 \cdot 0}{3}))$

Step 3.1.3

Factor $3$ out of $3 (50) + 3 (120 \cdot 0)$ .

$5 (\cos (\frac{3 (50 + 120 \cdot 0)}{3}) + i \sin (\frac{150 + 360 \cdot 0}{3}))$

Step 3.1.4

Cancel the common factors.

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Step 3.1.4.1

Factor $3$ out of $3$ .

$5 (\cos (\frac{3 (50 + 120 \cdot 0)}{3 (1)}) + i \sin (\frac{150 + 360 \cdot 0}{3}))$

Step 3.1.4.2

Cancel the common factor.

$5 (\cos (\frac{3 (50 + 120 \cdot 0)}{3 \cdot 1}) + i \sin (\frac{150 + 360 \cdot 0}{3}))$

Step 3.1.4.3

Rewrite the expression.

$5 (\cos (\frac{50 + 120 \cdot 0}{1}) + i \sin (\frac{150 + 360 \cdot 0}{3}))$

Step 3.1.4.4

Divide $50 + 120 \cdot 0$ by $1$ .

$5 (\cos (50 + 120 \cdot 0) + i \sin (\frac{150 + 360 \cdot 0}{3}))$

Step 3.2

Multiply $120$ by $0$ .

$5 (\cos (50 + 0) + i \sin (\frac{150 + 360 \cdot 0}{3}))$

Step 3.3

Add $50$ and $0$ .

$5 (\cos (50) + i \sin (\frac{150 + 360 \cdot 0}{3}))$

Step 3.4

Cancel the common factor of $150 + 360 \cdot 0$ and $3$ .

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Step 3.4.1

Factor $3$ out of $150$ .

$5 (\cos (50) + i \sin (\frac{3 \cdot 50 + 360 \cdot 0}{3}))$

Step 3.4.2

Factor $3$ out of $360 \cdot 0$ .

$5 (\cos (50) + i \sin (\frac{3 \cdot 50 + 3 (120 \cdot 0)}{3}))$

Step 3.4.3

Factor $3$ out of $3 (50) + 3 (120 \cdot 0)$ .

$5 (\cos (50) + i \sin (\frac{3 (50 + 120 \cdot 0)}{3}))$

Step 3.4.4

Cancel the common factors.

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Step 3.4.4.1

Factor $3$ out of $3$ .

$5 (\cos (50) + i \sin (\frac{3 (50 + 120 \cdot 0)}{3 (1)}))$

Step 3.4.4.2

Cancel the common factor.

$5 (\cos (50) + i \sin (\frac{3 (50 + 120 \cdot 0)}{3 \cdot 1}))$

Step 3.4.4.3

Rewrite the expression.

$5 (\cos (50) + i \sin (\frac{50 + 120 \cdot 0}{1}))$

Step 3.4.4.4

Divide $50 + 120 \cdot 0$ by $1$ .

$5 (\cos (50) + i \sin (50 + 120 \cdot 0))$

Step 3.5

Multiply $120$ by $0$ .

$5 (\cos (50) + i \sin (50 + 0))$

Step 3.6

Add $50$ and $0$ .

$5 (\cos (50) + i \sin (50))$

Step 4

Apply the distributive property.

$5 \cos (50) + 5 i \sin (50)$