Solve for z |(i+2 square root of 2)*z|=6

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

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.

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

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

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

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

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

Simplify the left side.

Multiply the numerator and denominator of $\frac{((i + 2 \sqrt{2}) \cdot z)}{2.82842712 + 1 i}$ by the conjugate of $2.82842712 + 1 i$ to make the denominator real.

$\frac{((i + 2 \sqrt{2}) \cdot z)}{2.82842712 + 1 i} \cdot \frac{2.82842712 - i}{2.82842712 - i} = \frac{6}{i + 2 \sqrt{2}}$

Multiply.

Combine.

$\frac{((i + 2 \sqrt{2}) \cdot z) (2.82842712 - i)}{(2.82842712 + 1 i) (2.82842712 - i)} = \frac{6}{i + 2 \sqrt{2}}$

Simplify the numerator.

Apply the distributive property.

$\frac{(i z + 2 \sqrt{2} z) (2.82842712 - i)}{(2.82842712 + 1 i) (2.82842712 - i)} = \frac{6}{i + 2 \sqrt{2}}$

Apply the distributive property.

$\frac{(i z + 2 \sqrt{2} z) \cdot 2.82842712 + (i z + 2 \sqrt{2} z) (- i)}{(2.82842712 + 1 i) (2.82842712 - i)} = \frac{6}{i + 2 \sqrt{2}}$

Simplify each term.

Apply the distributive property.

$\frac{i z \cdot 2.82842712 + 2 \sqrt{2} z \cdot 2.82842712 + (i z + 2 \sqrt{2} z) (- i)}{(2.82842712 + 1 i) (2.82842712 - i)} = \frac{6}{i + 2 \sqrt{2}}$

Move $2.82842712$ to the left of $i z$ .

$\frac{2.82842712 \cdot (i z) + 2 \sqrt{2} z \cdot 2.82842712 + (i z + 2 \sqrt{2} z) (- i)}{(2.82842712 + 1 i) (2.82842712 - i)} = \frac{6}{i + 2 \sqrt{2}}$

Multiply $2 \sqrt{2} z \cdot 2.82842712$ .

Multiply $2.82842712$ by $2$ .

$\frac{2.82842712 \cdot (i z) + 5.65685424 \sqrt{2} z + (i z + 2 \sqrt{2} z) (- i)}{(2.82842712 + 1 i) (2.82842712 - i)} = \frac{6}{i + 2 \sqrt{2}}$

Multiply $5.65685424$ by $\sqrt{2}$ .

$\frac{2.82842712 \cdot (i z) + 8 z + (i z + 2 \sqrt{2} z) (- i)}{(2.82842712 + 1 i) (2.82842712 - i)} = \frac{6}{i + 2 \sqrt{2}}$

Remove parentheses.

$\frac{2.82842712 i z + 8 z + (i z + 2 \sqrt{2} z) (- i)}{(2.82842712 + 1 i) (2.82842712 - i)} = \frac{6}{i + 2 \sqrt{2}}$

Apply the distributive property.

$\frac{2.82842712 i z + 8 z + i z (- i) + 2 \sqrt{2} z (- i)}{(2.82842712 + 1 i) (2.82842712 - i)} = \frac{6}{i + 2 \sqrt{2}}$

Multiply $i z (- i)$ .

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

$\frac{2.82842712 i z + 8 z + z (- (i^{1} i)) + 2 \sqrt{2} z (- i)}{(2.82842712 + 1 i) (2.82842712 - i)} = \frac{6}{i + 2 \sqrt{2}}$

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

$\frac{2.82842712 i z + 8 z + z (- (i^{1} i^{1})) + 2 \sqrt{2} z (- i)}{(2.82842712 + 1 i) (2.82842712 - i)} = \frac{6}{i + 2 \sqrt{2}}$

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

$\frac{2.82842712 i z + 8 z + z (- i^{1 + 1}) + 2 \sqrt{2} z (- i)}{(2.82842712 + 1 i) (2.82842712 - i)} = \frac{6}{i + 2 \sqrt{2}}$

Add $1$ and $1$ .

$\frac{2.82842712 i z + 8 z + z (- i^{2}) + 2 \sqrt{2} z (- i)}{(2.82842712 + 1 i) (2.82842712 - i)} = \frac{6}{i + 2 \sqrt{2}}$

Multiply $- 1$ by $2$ .

$\frac{2.82842712 i z + 8 z + z (- i^{2}) - 2 \sqrt{2} z i}{(2.82842712 + 1 i) (2.82842712 - i)} = \frac{6}{i + 2 \sqrt{2}}$

Simplify each term.

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

$\frac{2.82842712 i z + 8 z + z (- - 1) - 2 \sqrt{2} z i}{(2.82842712 + 1 i) (2.82842712 - i)} = \frac{6}{i + 2 \sqrt{2}}$

Multiply $- 1$ by $- 1$ .

$\frac{2.82842712 i z + 8 z + z \cdot 1 - 2 \sqrt{2} z i}{(2.82842712 + 1 i) (2.82842712 - i)} = \frac{6}{i + 2 \sqrt{2}}$

Multiply $z$ by $1$ .

$\frac{2.82842712 i z + 8 z + z - 2 \sqrt{2} z i}{(2.82842712 + 1 i) (2.82842712 - i)} = \frac{6}{i + 2 \sqrt{2}}$

Add $8 z$ and $z$ .

$\frac{2.82842712 i z + 9 z - 2 \sqrt{2} z i}{(2.82842712 + 1 i) (2.82842712 - i)} = \frac{6}{i + 2 \sqrt{2}}$

Simplify the denominator.

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

Apply the distributive property.

$\frac{2.82842712 i z + 9 z - 2 \sqrt{2} z i}{2.82842712 (2.82842712 - i) + 1 i (2.82842712 - i)} = \frac{6}{i + 2 \sqrt{2}}$

Apply the distributive property.

$\frac{2.82842712 i z + 9 z - 2 \sqrt{2} z i}{2.82842712 \cdot 2.82842712 + 2.82842712 (- i) + 1 i (2.82842712 - i)} = \frac{6}{i + 2 \sqrt{2}}$

Apply the distributive property.

$\frac{2.82842712 i z + 9 z - 2 \sqrt{2} z i}{2.82842712 \cdot 2.82842712 + 2.82842712 (- i) + 1 i \cdot 2.82842712 + 1 i (- i)} = \frac{6}{i + 2 \sqrt{2}}$

Simplify.

Multiply $2.82842712$ by $2.82842712$ .

$\frac{2.82842712 i z + 9 z - 2 \sqrt{2} z i}{8 + 2.82842712 (- i) + 1 i \cdot 2.82842712 + 1 i (- i)} = \frac{6}{i + 2 \sqrt{2}}$

Multiply $- 1$ by $2.82842712$ .

$\frac{2.82842712 i z + 9 z - 2 \sqrt{2} z i}{8 - 2.82842712 i + 1 i \cdot 2.82842712 + 1 i (- i)} = \frac{6}{i + 2 \sqrt{2}}$

Multiply $2.82842712$ by $1$ .

$\frac{2.82842712 i z + 9 z - 2 \sqrt{2} z i}{8 - 2.82842712 i + 2.82842712 i + 1 i (- i)} = \frac{6}{i + 2 \sqrt{2}}$

Multiply $- 1$ by $1$ .

$\frac{2.82842712 i z + 9 z - 2 \sqrt{2} z i}{8 - 2.82842712 i + 2.82842712 i - i i} = \frac{6}{i + 2 \sqrt{2}}$

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

$\frac{2.82842712 i z + 9 z - 2 \sqrt{2} z i}{8 - 2.82842712 i + 2.82842712 i - (i^{1} i)} = \frac{6}{i + 2 \sqrt{2}}$

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

$\frac{2.82842712 i z + 9 z - 2 \sqrt{2} z i}{8 - 2.82842712 i + 2.82842712 i - (i^{1} i^{1})} = \frac{6}{i + 2 \sqrt{2}}$

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

$\frac{2.82842712 i z + 9 z - 2 \sqrt{2} z i}{8 - 2.82842712 i + 2.82842712 i - i^{1 + 1}} = \frac{6}{i + 2 \sqrt{2}}$

Add $1$ and $1$ .

$\frac{2.82842712 i z + 9 z - 2 \sqrt{2} z i}{8 - 2.82842712 i + 2.82842712 i - i^{2}} = \frac{6}{i + 2 \sqrt{2}}$

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

$\frac{2.82842712 i z + 9 z - 2 \sqrt{2} z i}{8 + 0 i - i^{2}} = \frac{6}{i + 2 \sqrt{2}}$

Simplify each term.

Multiply $0$ by $i$ .

$\frac{2.82842712 i z + 9 z - 2 \sqrt{2} z i}{8 + 0 - i^{2}} = \frac{6}{i + 2 \sqrt{2}}$

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

$\frac{2.82842712 i z + 9 z - 2 \sqrt{2} z i}{8 + 0 - - 1} = \frac{6}{i + 2 \sqrt{2}}$

Multiply $- 1$ by $- 1$ .

$\frac{2.82842712 i z + 9 z - 2 \sqrt{2} z i}{8 + 0 + 1} = \frac{6}{i + 2 \sqrt{2}}$

Add $8$ and $0$ .

$\frac{2.82842712 i z + 9 z - 2 \sqrt{2} z i}{8 + 1} = \frac{6}{i + 2 \sqrt{2}}$

Add $8$ and $1$ .

$\frac{2.82842712 i z + 9 z - 2 \sqrt{2} z i}{9} = \frac{6}{i + 2 \sqrt{2}}$

Reorder terms.

$\frac{2.82842712 i z - 2 i z \sqrt{2} + 9 z}{9} = \frac{6}{i + 2 \sqrt{2}}$

Factor $1$ out of $2.82842712 i z$ .

$\frac{1 (2.82842712 i z) - 2 i z \sqrt{2} + 9 z}{9} = \frac{6}{i + 2 \sqrt{2}}$

Factor $1$ out of $- 2 i z \sqrt{2}$ .

$\frac{1 (2.82842712 i z) + 1 (- 2 i z \sqrt{2}) + 9 z}{9} = \frac{6}{i + 2 \sqrt{2}}$

Factor $1$ out of $1 (2.82842712 i z) + 1 (- 2 i z \sqrt{2})$ .

$\frac{1 (2.82842712 i z - 2 i z \sqrt{2}) + 9 z}{9} = \frac{6}{i + 2 \sqrt{2}}$

Factor $1$ out of $9 z$ .

$\frac{1 (2.82842712 i z - 2 i z \sqrt{2}) + 1 (9 z)}{9} = \frac{6}{i + 2 \sqrt{2}}$

Factor $1$ out of $1 (2.82842712 i z - 2 i z \sqrt{2}) + 1 (9 z)$ .

$\frac{1 (2.82842712 i z - 2 i z \sqrt{2} + 9 z)}{9} = \frac{6}{i + 2 \sqrt{2}}$

Factor $9$ out of $9$ .

$\frac{1 (2.82842712 i z - 2 i z \sqrt{2} + 9 z)}{9 (1)} = \frac{6}{i + 2 \sqrt{2}}$

Separate fractions.

$\frac{1}{9} \cdot \frac{2.82842712 i z - 2 i z \sqrt{2} + 9 z}{1} = \frac{6}{i + 2 \sqrt{2}}$

Simplify the expression.

Divide $1$ by $9$ .

$0.11111111 \frac{2.82842712 i z - 2 i z \sqrt{2} + 9 z}{1} = \frac{6}{i + 2 \sqrt{2}}$

Divide $2.82842712 i z - 2 i z \sqrt{2} + 9 z$ by $1$ .

$0.11111111 (2.82842712 i z - 2 i z \sqrt{2} + 9 z) = \frac{6}{i + 2 \sqrt{2}}$

Apply the distributive property.

$0.11111111 (2.82842712 i z) + 0.11111111 (- 2 i z \sqrt{2}) + 0.11111111 (9 z) = \frac{6}{i + 2 \sqrt{2}}$

Simplify.

Multiply $2.82842712$ by $0.11111111$ .

$0.31426968 (i z) + 0.11111111 (- 2 i z \sqrt{2}) + 0.11111111 (9 z) = \frac{6}{i + 2 \sqrt{2}}$

Multiply $0.11111111 (- 2 i z \sqrt{2})$ .

Multiply $- 2$ by $0.11111111$ .

$0.31426968 (i z) - 0.22222222 (i z \sqrt{2}) + 0.11111111 (9 z) = \frac{6}{i + 2 \sqrt{2}}$

Multiply $\sqrt{2}$ by $- 0.22222222$ .

$0.31426968 (i z) - 0.31426968 (i z) + 0.11111111 (9 z) = \frac{6}{i + 2 \sqrt{2}}$

Multiply $0.11111111 (9 z)$ .

Multiply $9$ by $0.11111111$ .

$0.31426968 (i z) - 0.31426968 (i z) + 1 z = \frac{6}{i + 2 \sqrt{2}}$

Multiply $z$ by $1$ .

$0.31426968 (i z) - 0.31426968 (i z) + z = \frac{6}{i + 2 \sqrt{2}}$

Subtract $0.31426968 (i z)$ from $0.31426968 (i z)$ .

$0 i z + z = \frac{6}{i + 2 \sqrt{2}}$

Simplify each term.

Multiply $0$ by $i$ .

$0 z + z = \frac{6}{i + 2 \sqrt{2}}$

Multiply $0$ by $z$ .

$0 + z = \frac{6}{i + 2 \sqrt{2}}$

Add $0$ and $z$ .

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

Simplify the right side.

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.

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

Multiply.

Combine.

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

Simplify the numerator.

Apply the distributive property.

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

Multiply $6$ by $2.82842712$ .

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

Multiply $- 1$ by $6$ .

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

Simplify the denominator.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify.

Multiply $2.82842712$ by $2.82842712$ .

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

Multiply $- 1$ by $2.82842712$ .

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

Multiply $2.82842712$ by $1$ .

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

Multiply $- 1$ by $1$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

Simplify each term.

Multiply $0$ by $i$ .

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

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

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

Multiply $- 1$ by $- 1$ .

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

Add $8$ and $0$ .

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

Add $8$ and $1$ .

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

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

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

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

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

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

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

Factor $9$ out of $9$ .

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

Separate fractions.

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

Simplify the expression.

Divide $1$ by $9$ .

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

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

$z = 0.11111111 (16.97056274 - 6 i)$

Apply the distributive property.

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

Multiply.

Multiply $0.11111111$ by $16.97056274$ .

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

Multiply $- 6$ by $0.11111111$ .

$z = 1.88561808 - 0.66666666 i$

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

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

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

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

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

Simplify the left side.

Multiply the numerator and denominator of $\frac{((i + 2 \sqrt{2}) \cdot z)}{2.82842712 + 1 i}$ by the conjugate of $2.82842712 + 1 i$ to make the denominator real.

$\frac{((i + 2 \sqrt{2}) \cdot z)}{2.82842712 + 1 i} \cdot \frac{2.82842712 - i}{2.82842712 - i} = \frac{- 6}{i + 2 \sqrt{2}}$

Multiply.

Combine.

$\frac{((i + 2 \sqrt{2}) \cdot z) (2.82842712 - i)}{(2.82842712 + 1 i) (2.82842712 - i)} = \frac{- 6}{i + 2 \sqrt{2}}$

Simplify the numerator.

Apply the distributive property.

$\frac{(i z + 2 \sqrt{2} z) (2.82842712 - i)}{(2.82842712 + 1 i) (2.82842712 - i)} = \frac{- 6}{i + 2 \sqrt{2}}$

Apply the distributive property.

$\frac{(i z + 2 \sqrt{2} z) \cdot 2.82842712 + (i z + 2 \sqrt{2} z) (- i)}{(2.82842712 + 1 i) (2.82842712 - i)} = \frac{- 6}{i + 2 \sqrt{2}}$

Simplify each term.

Apply the distributive property.

$\frac{i z \cdot 2.82842712 + 2 \sqrt{2} z \cdot 2.82842712 + (i z + 2 \sqrt{2} z) (- i)}{(2.82842712 + 1 i) (2.82842712 - i)} = \frac{- 6}{i + 2 \sqrt{2}}$

Move $2.82842712$ to the left of $i z$ .

$\frac{2.82842712 \cdot (i z) + 2 \sqrt{2} z \cdot 2.82842712 + (i z + 2 \sqrt{2} z) (- i)}{(2.82842712 + 1 i) (2.82842712 - i)} = \frac{- 6}{i + 2 \sqrt{2}}$

Multiply $2 \sqrt{2} z \cdot 2.82842712$ .

Multiply $2.82842712$ by $2$ .

$\frac{2.82842712 \cdot (i z) + 5.65685424 \sqrt{2} z + (i z + 2 \sqrt{2} z) (- i)}{(2.82842712 + 1 i) (2.82842712 - i)} = \frac{- 6}{i + 2 \sqrt{2}}$

Multiply $5.65685424$ by $\sqrt{2}$ .

$\frac{2.82842712 \cdot (i z) + 8 z + (i z + 2 \sqrt{2} z) (- i)}{(2.82842712 + 1 i) (2.82842712 - i)} = \frac{- 6}{i + 2 \sqrt{2}}$

Remove parentheses.

$\frac{2.82842712 i z + 8 z + (i z + 2 \sqrt{2} z) (- i)}{(2.82842712 + 1 i) (2.82842712 - i)} = \frac{- 6}{i + 2 \sqrt{2}}$

Apply the distributive property.

$\frac{2.82842712 i z + 8 z + i z (- i) + 2 \sqrt{2} z (- i)}{(2.82842712 + 1 i) (2.82842712 - i)} = \frac{- 6}{i + 2 \sqrt{2}}$

Multiply $i z (- i)$ .

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

$\frac{2.82842712 i z + 8 z + z (- (i^{1} i)) + 2 \sqrt{2} z (- i)}{(2.82842712 + 1 i) (2.82842712 - i)} = \frac{- 6}{i + 2 \sqrt{2}}$

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

$\frac{2.82842712 i z + 8 z + z (- (i^{1} i^{1})) + 2 \sqrt{2} z (- i)}{(2.82842712 + 1 i) (2.82842712 - i)} = \frac{- 6}{i + 2 \sqrt{2}}$

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

$\frac{2.82842712 i z + 8 z + z (- i^{1 + 1}) + 2 \sqrt{2} z (- i)}{(2.82842712 + 1 i) (2.82842712 - i)} = \frac{- 6}{i + 2 \sqrt{2}}$

Add $1$ and $1$ .

$\frac{2.82842712 i z + 8 z + z (- i^{2}) + 2 \sqrt{2} z (- i)}{(2.82842712 + 1 i) (2.82842712 - i)} = \frac{- 6}{i + 2 \sqrt{2}}$

Multiply $- 1$ by $2$ .

$\frac{2.82842712 i z + 8 z + z (- i^{2}) - 2 \sqrt{2} z i}{(2.82842712 + 1 i) (2.82842712 - i)} = \frac{- 6}{i + 2 \sqrt{2}}$

Simplify each term.

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

$\frac{2.82842712 i z + 8 z + z (- - 1) - 2 \sqrt{2} z i}{(2.82842712 + 1 i) (2.82842712 - i)} = \frac{- 6}{i + 2 \sqrt{2}}$

Multiply $- 1$ by $- 1$ .

$\frac{2.82842712 i z + 8 z + z \cdot 1 - 2 \sqrt{2} z i}{(2.82842712 + 1 i) (2.82842712 - i)} = \frac{- 6}{i + 2 \sqrt{2}}$

Multiply $z$ by $1$ .

$\frac{2.82842712 i z + 8 z + z - 2 \sqrt{2} z i}{(2.82842712 + 1 i) (2.82842712 - i)} = \frac{- 6}{i + 2 \sqrt{2}}$

Add $8 z$ and $z$ .

$\frac{2.82842712 i z + 9 z - 2 \sqrt{2} z i}{(2.82842712 + 1 i) (2.82842712 - i)} = \frac{- 6}{i + 2 \sqrt{2}}$

Simplify the denominator.

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

Apply the distributive property.

$\frac{2.82842712 i z + 9 z - 2 \sqrt{2} z i}{2.82842712 (2.82842712 - i) + 1 i (2.82842712 - i)} = \frac{- 6}{i + 2 \sqrt{2}}$

Apply the distributive property.

$\frac{2.82842712 i z + 9 z - 2 \sqrt{2} z i}{2.82842712 \cdot 2.82842712 + 2.82842712 (- i) + 1 i (2.82842712 - i)} = \frac{- 6}{i + 2 \sqrt{2}}$

Apply the distributive property.

$\frac{2.82842712 i z + 9 z - 2 \sqrt{2} z i}{2.82842712 \cdot 2.82842712 + 2.82842712 (- i) + 1 i \cdot 2.82842712 + 1 i (- i)} = \frac{- 6}{i + 2 \sqrt{2}}$

Simplify.

Multiply $2.82842712$ by $2.82842712$ .

$\frac{2.82842712 i z + 9 z - 2 \sqrt{2} z i}{8 + 2.82842712 (- i) + 1 i \cdot 2.82842712 + 1 i (- i)} = \frac{- 6}{i + 2 \sqrt{2}}$

Multiply $- 1$ by $2.82842712$ .

$\frac{2.82842712 i z + 9 z - 2 \sqrt{2} z i}{8 - 2.82842712 i + 1 i \cdot 2.82842712 + 1 i (- i)} = \frac{- 6}{i + 2 \sqrt{2}}$

Multiply $2.82842712$ by $1$ .

$\frac{2.82842712 i z + 9 z - 2 \sqrt{2} z i}{8 - 2.82842712 i + 2.82842712 i + 1 i (- i)} = \frac{- 6}{i + 2 \sqrt{2}}$

Multiply $- 1$ by $1$ .

$\frac{2.82842712 i z + 9 z - 2 \sqrt{2} z i}{8 - 2.82842712 i + 2.82842712 i - i i} = \frac{- 6}{i + 2 \sqrt{2}}$

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

$\frac{2.82842712 i z + 9 z - 2 \sqrt{2} z i}{8 - 2.82842712 i + 2.82842712 i - (i^{1} i)} = \frac{- 6}{i + 2 \sqrt{2}}$

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

$\frac{2.82842712 i z + 9 z - 2 \sqrt{2} z i}{8 - 2.82842712 i + 2.82842712 i - (i^{1} i^{1})} = \frac{- 6}{i + 2 \sqrt{2}}$

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

$\frac{2.82842712 i z + 9 z - 2 \sqrt{2} z i}{8 - 2.82842712 i + 2.82842712 i - i^{1 + 1}} = \frac{- 6}{i + 2 \sqrt{2}}$

Add $1$ and $1$ .

$\frac{2.82842712 i z + 9 z - 2 \sqrt{2} z i}{8 - 2.82842712 i + 2.82842712 i - i^{2}} = \frac{- 6}{i + 2 \sqrt{2}}$

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

$\frac{2.82842712 i z + 9 z - 2 \sqrt{2} z i}{8 + 0 i - i^{2}} = \frac{- 6}{i + 2 \sqrt{2}}$

Simplify each term.

Multiply $0$ by $i$ .

$\frac{2.82842712 i z + 9 z - 2 \sqrt{2} z i}{8 + 0 - i^{2}} = \frac{- 6}{i + 2 \sqrt{2}}$

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

$\frac{2.82842712 i z + 9 z - 2 \sqrt{2} z i}{8 + 0 - - 1} = \frac{- 6}{i + 2 \sqrt{2}}$

Multiply $- 1$ by $- 1$ .

$\frac{2.82842712 i z + 9 z - 2 \sqrt{2} z i}{8 + 0 + 1} = \frac{- 6}{i + 2 \sqrt{2}}$

Add $8$ and $0$ .

$\frac{2.82842712 i z + 9 z - 2 \sqrt{2} z i}{8 + 1} = \frac{- 6}{i + 2 \sqrt{2}}$

Add $8$ and $1$ .

$\frac{2.82842712 i z + 9 z - 2 \sqrt{2} z i}{9} = \frac{- 6}{i + 2 \sqrt{2}}$

Reorder terms.

$\frac{2.82842712 i z - 2 i z \sqrt{2} + 9 z}{9} = \frac{- 6}{i + 2 \sqrt{2}}$

Factor $1$ out of $2.82842712 i z$ .

$\frac{1 (2.82842712 i z) - 2 i z \sqrt{2} + 9 z}{9} = \frac{- 6}{i + 2 \sqrt{2}}$

Factor $1$ out of $- 2 i z \sqrt{2}$ .

$\frac{1 (2.82842712 i z) + 1 (- 2 i z \sqrt{2}) + 9 z}{9} = \frac{- 6}{i + 2 \sqrt{2}}$

Factor $1$ out of $1 (2.82842712 i z) + 1 (- 2 i z \sqrt{2})$ .

$\frac{1 (2.82842712 i z - 2 i z \sqrt{2}) + 9 z}{9} = \frac{- 6}{i + 2 \sqrt{2}}$

Factor $1$ out of $9 z$ .

$\frac{1 (2.82842712 i z - 2 i z \sqrt{2}) + 1 (9 z)}{9} = \frac{- 6}{i + 2 \sqrt{2}}$

Factor $1$ out of $1 (2.82842712 i z - 2 i z \sqrt{2}) + 1 (9 z)$ .

$\frac{1 (2.82842712 i z - 2 i z \sqrt{2} + 9 z)}{9} = \frac{- 6}{i + 2 \sqrt{2}}$

Factor $9$ out of $9$ .

$\frac{1 (2.82842712 i z - 2 i z \sqrt{2} + 9 z)}{9 (1)} = \frac{- 6}{i + 2 \sqrt{2}}$

Separate fractions.

$\frac{1}{9} \cdot \frac{2.82842712 i z - 2 i z \sqrt{2} + 9 z}{1} = \frac{- 6}{i + 2 \sqrt{2}}$

Simplify the expression.

Divide $1$ by $9$ .

$0.11111111 \frac{2.82842712 i z - 2 i z \sqrt{2} + 9 z}{1} = \frac{- 6}{i + 2 \sqrt{2}}$

Divide $2.82842712 i z - 2 i z \sqrt{2} + 9 z$ by $1$ .

$0.11111111 (2.82842712 i z - 2 i z \sqrt{2} + 9 z) = \frac{- 6}{i + 2 \sqrt{2}}$

Apply the distributive property.

$0.11111111 (2.82842712 i z) + 0.11111111 (- 2 i z \sqrt{2}) + 0.11111111 (9 z) = \frac{- 6}{i + 2 \sqrt{2}}$

Simplify.

Multiply $2.82842712$ by $0.11111111$ .

$0.31426968 (i z) + 0.11111111 (- 2 i z \sqrt{2}) + 0.11111111 (9 z) = \frac{- 6}{i + 2 \sqrt{2}}$

Multiply $0.11111111 (- 2 i z \sqrt{2})$ .

Multiply $- 2$ by $0.11111111$ .

$0.31426968 (i z) - 0.22222222 (i z \sqrt{2}) + 0.11111111 (9 z) = \frac{- 6}{i + 2 \sqrt{2}}$

Multiply $\sqrt{2}$ by $- 0.22222222$ .

$0.31426968 (i z) - 0.31426968 (i z) + 0.11111111 (9 z) = \frac{- 6}{i + 2 \sqrt{2}}$

Multiply $0.11111111 (9 z)$ .

Multiply $9$ by $0.11111111$ .

$0.31426968 (i z) - 0.31426968 (i z) + 1 z = \frac{- 6}{i + 2 \sqrt{2}}$

Multiply $z$ by $1$ .

$0.31426968 (i z) - 0.31426968 (i z) + z = \frac{- 6}{i + 2 \sqrt{2}}$

Subtract $0.31426968 (i z)$ from $0.31426968 (i z)$ .

$0 i z + z = \frac{- 6}{i + 2 \sqrt{2}}$

Simplify each term.

Multiply $0$ by $i$ .

$0 z + z = \frac{- 6}{i + 2 \sqrt{2}}$

Multiply $0$ by $z$ .

$0 + z = \frac{- 6}{i + 2 \sqrt{2}}$

Add $0$ and $z$ .

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

Simplify the right side.

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.

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

Multiply.

Combine.

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

Simplify the numerator.

Apply the distributive property.

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

Multiply $- 6$ by $2.82842712$ .

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

Multiply $- 1$ by $- 6$ .

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

Simplify the denominator.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify.

Multiply $2.82842712$ by $2.82842712$ .

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

Multiply $- 1$ by $2.82842712$ .

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

Multiply $2.82842712$ by $1$ .

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

Multiply $- 1$ by $1$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

Simplify each term.

Multiply $0$ by $i$ .

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

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

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

Multiply $- 1$ by $- 1$ .

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

Add $8$ and $0$ .

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

Add $8$ and $1$ .

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

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

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

Factor $1$ out of $6 i$ .

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

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

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

Factor $9$ out of $9$ .

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

Separate fractions.

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

Simplify the expression.

Divide $1$ by $9$ .

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

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

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

Apply the distributive property.

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

Multiply.

Multiply $0.11111111$ by $- 16.97056274$ .

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

Multiply $6$ by $0.11111111$ .

$z = - 1.88561808 + 0.66666666 i$

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

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

Solve for z |(i+2 square root of 2)*z|=6

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