Trigonometry Examples

$| 2 - 2 \sqrt{3} i |$

Step 1

Use the formula $| a + b i | = \sqrt{a^{2} + b^{2}}$ to find the magnitude.

$\sqrt{2^{2} + {(- 2 \sqrt{3})}^{2}}$

Step 2

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

$\sqrt{4 + {(- 2 \sqrt{3})}^{2}}$

Step 3

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

$\sqrt{4 + {(- 2)}^{2} {\sqrt{3}}^{2}}$

Step 4

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

$\sqrt{4 + 4 {\sqrt{3}}^{2}}$

Step 5

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 5.1

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

$\sqrt{4 + 4 {(3^{\frac{1}{2}})}^{2}}$

Step 5.2

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

$\sqrt{4 + 4 \cdot 3^{\frac{1}{2} \cdot 2}}$

Step 5.3

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

$\sqrt{4 + 4 \cdot 3^{\frac{2}{2}}}$

Step 5.4

Cancel the common factor of $2$ .

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Step 5.4.1

Cancel the common factor.

$\sqrt{4 + 4 \cdot 3^{\frac{2}{2}}}$

Step 5.4.2

Rewrite the expression.

$\sqrt{4 + 4 \cdot 3^{1}}$

Step 5.5

Evaluate the exponent.

$\sqrt{4 + 4 \cdot 3}$

Step 6

Multiply $4$ by $3$ .

$\sqrt{4 + 12}$

Step 7

Add $4$ and $12$ .

$\sqrt{16}$

Step 8

Rewrite $16$ as $4^{2}$ .

$\sqrt{4^{2}}$

Step 9

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

$4$

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