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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