Precalculus Examples

$| z | = | - 4 + 4 i |$

Step 1

Simplify $| - 4 + 4 i |$ .

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Step 1.1

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

$| z | = \sqrt{{(- 4)}^{2} + 4^{2}}$

Step 1.2

Raise $- 4$ to the power of $2$ .

$| z | = \sqrt{16 + 4^{2}}$

Step 1.3

Raise $4$ to the power of $2$ .

$| z | = \sqrt{16 + 16}$

Step 1.4

Add $16$ and $16$ .

$| z | = \sqrt{32}$

Step 1.5

Rewrite $32$ as $4^{2} \cdot 2$ .

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Step 1.5.1

Factor $16$ out of $32$ .

$| z | = \sqrt{16 (2)}$

Step 1.5.2

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

$| z | = \sqrt{4^{2} \cdot 2}$

Step 1.6

Pull terms out from under the radical.

$| z | = 4 \sqrt{2}$

Step 2

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$z = \pm 4 \sqrt{2}$

Step 3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.1

First, use the positive value of the $\pm$ to find the first solution.

$z = 4 \sqrt{2}$

Step 3.2

Next, use the negative value of the $\pm$ to find the second solution.

$z = - 4 \sqrt{2}$

Step 3.3

The complete solution is the result of both the positive and negative portions of the solution.

$z = 4 \sqrt{2}, - 4 \sqrt{2}$

Step 4

Exclude the solutions that do not make $| z | = | - 4 + 4 i |$ true.

$No solution$