Basic Math Examples

$| (i + 2 \sqrt{2}) \cdot z | = 6$

Step 1

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

$(i + 2 \sqrt{2}) \cdot z = \pm 6$

Step 2

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

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Step 2.1

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

$(i + 2 \sqrt{2}) \cdot z = 6$

Step 2.2

Divide each term in $(i + 2 \sqrt{2}) \cdot z = 6$ by $i + 2 \sqrt{2}$ and simplify.

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Step 2.2.1

Divide each term in $(i + 2 \sqrt{2}) \cdot z = 6$ by $i + 2 \sqrt{2}$ .

$\frac{(i + 2 \sqrt{2}) \cdot z}{i + 2 \sqrt{2}} = \frac{6}{i + 2 \sqrt{2}}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $i + 2 \sqrt{2}$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{(i + 2 \sqrt{2}) \cdot z}{i + 2 \sqrt{2}} = \frac{6}{i + 2 \sqrt{2}}$

Step 2.2.2.1.2

Divide $z$ by $1$ .

$z = \frac{6}{i + 2 \sqrt{2}}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Multiply the numerator and denominator of $\frac{6}{2.82842712 + 1 i}$ by the conjugate of $2.82842712 + 1 i$ to make the denominator real.

$z = \frac{6}{2.82842712 + 1 i} \cdot \frac{2.82842712 - i}{2.82842712 - i}$

Step 2.2.3.2

Multiply.

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Step 2.2.3.2.1

Combine.

$z = \frac{6 (2.82842712 - i)}{(2.82842712 + 1 i) (2.82842712 - i)}$

Step 2.2.3.2.2

Simplify the numerator.

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Step 2.2.3.2.2.1

Apply the distributive property.

$z = \frac{6 \cdot 2.82842712 + 6 (- i)}{(2.82842712 + 1 i) (2.82842712 - i)}$

Step 2.2.3.2.2.2

Multiply $6$ by $2.82842712$ .

$z = \frac{16.97056274 + 6 (- i)}{(2.82842712 + 1 i) (2.82842712 - i)}$

Step 2.2.3.2.2.3

Multiply $- 1$ by $6$ .

$z = \frac{16.97056274 - 6 i}{(2.82842712 + 1 i) (2.82842712 - i)}$

Step 2.2.3.2.3

Simplify the denominator.

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Step 2.2.3.2.3.1

Expand $(2.82842712 + 1 i) (2.82842712 - i)$ using the FOIL Method.

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Step 2.2.3.2.3.1.1

Apply the distributive property.

$z = \frac{16.97056274 - 6 i}{2.82842712 (2.82842712 - i) + 1 i (2.82842712 - i)}$

Step 2.2.3.2.3.1.2

Apply the distributive property.

$z = \frac{16.97056274 - 6 i}{2.82842712 \cdot 2.82842712 + 2.82842712 (- i) + 1 i (2.82842712 - i)}$

Step 2.2.3.2.3.1.3

Apply the distributive property.

$z = \frac{16.97056274 - 6 i}{2.82842712 \cdot 2.82842712 + 2.82842712 (- i) + 1 i \cdot 2.82842712 + 1 i (- i)}$

Step 2.2.3.2.3.2

Simplify.

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Step 2.2.3.2.3.2.1

Multiply $2.82842712$ by $2.82842712$ .

$z = \frac{16.97056274 - 6 i}{8 + 2.82842712 (- i) + 1 i \cdot 2.82842712 + 1 i (- i)}$

Step 2.2.3.2.3.2.2

Multiply $- 1$ by $2.82842712$ .

$z = \frac{16.97056274 - 6 i}{8 - 2.82842712 i + 1 i \cdot 2.82842712 + 1 i (- i)}$

Step 2.2.3.2.3.2.3

Multiply $2.82842712$ by $1$ .

$z = \frac{16.97056274 - 6 i}{8 - 2.82842712 i + 2.82842712 i + 1 i (- i)}$

Step 2.2.3.2.3.2.4

Multiply $- 1$ by $1$ .

$z = \frac{16.97056274 - 6 i}{8 - 2.82842712 i + 2.82842712 i - i i}$

Step 2.2.3.2.3.2.5

Raise $i$ to the power of $1$ .

$z = \frac{16.97056274 - 6 i}{8 - 2.82842712 i + 2.82842712 i - (i^{1} i)}$

Step 2.2.3.2.3.2.6

Raise $i$ to the power of $1$ .

$z = \frac{16.97056274 - 6 i}{8 - 2.82842712 i + 2.82842712 i - (i^{1} i^{1})}$

Step 2.2.3.2.3.2.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$z = \frac{16.97056274 - 6 i}{8 - 2.82842712 i + 2.82842712 i - i^{1 + 1}}$

Step 2.2.3.2.3.2.8

Add $1$ and $1$ .

$z = \frac{16.97056274 - 6 i}{8 - 2.82842712 i + 2.82842712 i - i^{2}}$

Step 2.2.3.2.3.2.9

Add $- 2.82842712 i$ and $2.82842712 i$ .

$z = \frac{16.97056274 - 6 i}{8 + 0 i - i^{2}}$

Step 2.2.3.2.3.3

Simplify each term.

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Step 2.2.3.2.3.3.1

Multiply $0$ by $i$ .

$z = \frac{16.97056274 - 6 i}{8 + 0 - i^{2}}$

Step 2.2.3.2.3.3.2

Rewrite $i^{2}$ as $- 1$ .

$z = \frac{16.97056274 - 6 i}{8 + 0 - - 1}$

Step 2.2.3.2.3.3.3

Multiply $- 1$ by $- 1$ .

$z = \frac{16.97056274 - 6 i}{8 + 0 + 1}$

Step 2.2.3.2.3.4

Add $8$ and $0$ .

$z = \frac{16.97056274 - 6 i}{8 + 1}$

Step 2.2.3.2.3.5

Add $8$ and $1$ .

$z = \frac{16.97056274 - 6 i}{9}$

Step 2.2.3.3

Rewrite $16.97056274$ as $1 (16.97056274)$ .

$z = \frac{1 (16.97056274) - 6 i}{9}$

Step 2.2.3.4

Factor $1$ out of $- 6 i$ .

$z = \frac{1 (16.97056274) + 1 (- 6 i)}{9}$

Step 2.2.3.5

Factor $1$ out of $1 (16.97056274) + 1 (- 6 i)$ .

$z = \frac{1 (16.97056274 - 6 i)}{9}$

Step 2.2.3.6

Factor $9$ out of $9$ .

$z = \frac{1 (16.97056274 - 6 i)}{9 (1)}$

Step 2.2.3.7

Separate fractions.

$z = \frac{1}{9} \cdot \frac{16.97056274 - 6 i}{1}$

Step 2.2.3.8

Simplify the expression.

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Step 2.2.3.8.1

Divide $1$ by $9$ .

$z = 0.11111111 \frac{16.97056274 - 6 i}{1}$

Step 2.2.3.8.2

Divide $16.97056274 - 6 i$ by $1$ .

$z = 0.11111111 (16.97056274 - 6 i)$

Step 2.2.3.9

Apply the distributive property.

$z = 0.11111111 \cdot 16.97056274 + 0.11111111 (- 6 i)$

Step 2.2.3.10

Multiply.

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Step 2.2.3.10.1

Multiply $0.11111111$ by $16.97056274$ .

$z = 1.88561808 + 0.11111111 (- 6 i)$

Step 2.2.3.10.2

Multiply $- 6$ by $0.11111111$ .

$z = 1.88561808 - 0.66666666 i$

Step 2.3

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

$(i + 2 \sqrt{2}) \cdot z = - 6$

Step 2.4

Divide each term in $(i + 2 \sqrt{2}) \cdot z = - 6$ by $i + 2 \sqrt{2}$ and simplify.

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Step 2.4.1

Divide each term in $(i + 2 \sqrt{2}) \cdot z = - 6$ by $i + 2 \sqrt{2}$ .

$\frac{(i + 2 \sqrt{2}) \cdot z}{i + 2 \sqrt{2}} = \frac{- 6}{i + 2 \sqrt{2}}$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Cancel the common factor of $i + 2 \sqrt{2}$ .

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Step 2.4.2.1.1

Cancel the common factor.

$\frac{(i + 2 \sqrt{2}) \cdot z}{i + 2 \sqrt{2}} = \frac{- 6}{i + 2 \sqrt{2}}$

Step 2.4.2.1.2

Divide $z$ by $1$ .

$z = \frac{- 6}{i + 2 \sqrt{2}}$

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Multiply the numerator and denominator of $\frac{- 6}{2.82842712 + 1 i}$ by the conjugate of $2.82842712 + 1 i$ to make the denominator real.

$z = \frac{- 6}{2.82842712 + 1 i} \cdot \frac{2.82842712 - i}{2.82842712 - i}$

Step 2.4.3.2

Multiply.

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Step 2.4.3.2.1

Combine.

$z = \frac{- 6 (2.82842712 - i)}{(2.82842712 + 1 i) (2.82842712 - i)}$

Step 2.4.3.2.2

Simplify the numerator.

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Step 2.4.3.2.2.1

Apply the distributive property.

$z = \frac{- 6 \cdot 2.82842712 - 6 (- i)}{(2.82842712 + 1 i) (2.82842712 - i)}$

Step 2.4.3.2.2.2

Multiply $- 6$ by $2.82842712$ .

$z = \frac{- 16.97056274 - 6 (- i)}{(2.82842712 + 1 i) (2.82842712 - i)}$

Step 2.4.3.2.2.3

Multiply $- 1$ by $- 6$ .

$z = \frac{- 16.97056274 + 6 i}{(2.82842712 + 1 i) (2.82842712 - i)}$

Step 2.4.3.2.3

Simplify the denominator.

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Step 2.4.3.2.3.1

Expand $(2.82842712 + 1 i) (2.82842712 - i)$ using the FOIL Method.

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Step 2.4.3.2.3.1.1

Apply the distributive property.

$z = \frac{- 16.97056274 + 6 i}{2.82842712 (2.82842712 - i) + 1 i (2.82842712 - i)}$

Step 2.4.3.2.3.1.2

Apply the distributive property.

$z = \frac{- 16.97056274 + 6 i}{2.82842712 \cdot 2.82842712 + 2.82842712 (- i) + 1 i (2.82842712 - i)}$

Step 2.4.3.2.3.1.3

Apply the distributive property.

$z = \frac{- 16.97056274 + 6 i}{2.82842712 \cdot 2.82842712 + 2.82842712 (- i) + 1 i \cdot 2.82842712 + 1 i (- i)}$

Step 2.4.3.2.3.2

Simplify.

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Step 2.4.3.2.3.2.1

Multiply $2.82842712$ by $2.82842712$ .

$z = \frac{- 16.97056274 + 6 i}{8 + 2.82842712 (- i) + 1 i \cdot 2.82842712 + 1 i (- i)}$

Step 2.4.3.2.3.2.2

Multiply $- 1$ by $2.82842712$ .

$z = \frac{- 16.97056274 + 6 i}{8 - 2.82842712 i + 1 i \cdot 2.82842712 + 1 i (- i)}$

Step 2.4.3.2.3.2.3

Multiply $2.82842712$ by $1$ .

$z = \frac{- 16.97056274 + 6 i}{8 - 2.82842712 i + 2.82842712 i + 1 i (- i)}$

Step 2.4.3.2.3.2.4

Multiply $- 1$ by $1$ .

$z = \frac{- 16.97056274 + 6 i}{8 - 2.82842712 i + 2.82842712 i - i i}$

Step 2.4.3.2.3.2.5

Raise $i$ to the power of $1$ .

$z = \frac{- 16.97056274 + 6 i}{8 - 2.82842712 i + 2.82842712 i - (i^{1} i)}$

Step 2.4.3.2.3.2.6

Raise $i$ to the power of $1$ .

$z = \frac{- 16.97056274 + 6 i}{8 - 2.82842712 i + 2.82842712 i - (i^{1} i^{1})}$

Step 2.4.3.2.3.2.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$z = \frac{- 16.97056274 + 6 i}{8 - 2.82842712 i + 2.82842712 i - i^{1 + 1}}$

Step 2.4.3.2.3.2.8

Add $1$ and $1$ .

$z = \frac{- 16.97056274 + 6 i}{8 - 2.82842712 i + 2.82842712 i - i^{2}}$

Step 2.4.3.2.3.2.9

Add $- 2.82842712 i$ and $2.82842712 i$ .

$z = \frac{- 16.97056274 + 6 i}{8 + 0 i - i^{2}}$

Step 2.4.3.2.3.3

Simplify each term.

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Step 2.4.3.2.3.3.1

Multiply $0$ by $i$ .

$z = \frac{- 16.97056274 + 6 i}{8 + 0 - i^{2}}$

Step 2.4.3.2.3.3.2

Rewrite $i^{2}$ as $- 1$ .

$z = \frac{- 16.97056274 + 6 i}{8 + 0 - - 1}$

Step 2.4.3.2.3.3.3

Multiply $- 1$ by $- 1$ .

$z = \frac{- 16.97056274 + 6 i}{8 + 0 + 1}$

Step 2.4.3.2.3.4

Add $8$ and $0$ .

$z = \frac{- 16.97056274 + 6 i}{8 + 1}$

Step 2.4.3.2.3.5

Add $8$ and $1$ .

$z = \frac{- 16.97056274 + 6 i}{9}$

Step 2.4.3.3

Rewrite $- 16.97056274$ as $1 (- 16.97056274)$ .

$z = \frac{1 (- 16.97056274) + 6 i}{9}$

Step 2.4.3.4

Factor $1$ out of $6 i$ .

$z = \frac{1 (- 16.97056274) + 1 (6 i)}{9}$

Step 2.4.3.5

Factor $1$ out of $1 (- 16.97056274) + 1 (6 i)$ .

$z = \frac{1 (- 16.97056274 + 6 i)}{9}$

Step 2.4.3.6

Factor $9$ out of $9$ .

$z = \frac{1 (- 16.97056274 + 6 i)}{9 (1)}$

Step 2.4.3.7

Separate fractions.

$z = \frac{1}{9} \cdot \frac{- 16.97056274 + 6 i}{1}$

Step 2.4.3.8

Simplify the expression.

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Step 2.4.3.8.1

Divide $1$ by $9$ .

$z = 0.11111111 \frac{- 16.97056274 + 6 i}{1}$

Step 2.4.3.8.2

Divide $- 16.97056274 + 6 i$ by $1$ .

$z = 0.11111111 (- 16.97056274 + 6 i)$

Step 2.4.3.9

Apply the distributive property.

$z = 0.11111111 \cdot - 16.97056274 + 0.11111111 (6 i)$

Step 2.4.3.10

Multiply.

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Step 2.4.3.10.1

Multiply $0.11111111$ by $- 16.97056274$ .

$z = - 1.88561808 + 0.11111111 (6 i)$

Step 2.4.3.10.2

Multiply $6$ by $0.11111111$ .

$z = - 1.88561808 + 0.66666666 i$

Step 2.5

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

$z = 1.88561808 - 0.66666666 i, - 1.88561808 + 0.66666666 i$