Solve for x n÷(( square root of 2)/2-( square root of 2i)/2)=nx

$n \div (\frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2}) = n x$

Rewrite the equation as $n x = n \div (\frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2})$ .

$n x = n \div (\frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2})$

Divide each term in $n x = n \div (\frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2})$ by $n$ and simplify.

Divide each term in $n x = n \div (\frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2})$ by $n$ .

$\frac{n x}{n} = \frac{n \div (\frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2})}{n}$

Simplify the left side.

Cancel the common factor of $n$ .

Cancel the common factor.

$\frac{n x}{n} = \frac{n \div (\frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2})}{n}$

Divide $x$ by $1$ .

$x = \frac{n \div (\frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2})}{n}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$x = n \div (\frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2}) \cdot \frac{1}{n}$

Rewrite the division as a fraction.

$x = \frac{n}{\frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2}} \cdot \frac{1}{n}$

Cancel the common factor of $n$ .

Cancel the common factor.

$x = \frac{n}{\frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2}} \cdot \frac{1}{n}$

Rewrite the expression.

$x = \frac{1}{\frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2}}$

Combine the numerators over the common denominator.

$x = \frac{1}{\frac{\sqrt{2} - \sqrt{2} i}{2}}$

Multiply the numerator by the reciprocal of the denominator.

$x = 1 \frac{2}{\sqrt{2} - \sqrt{2} i}$

Multiply $\frac{2}{\sqrt{2} - \sqrt{2} i}$ by $1$ .

$x = \frac{2}{\sqrt{2} - \sqrt{2} i}$

Multiply the numerator and denominator of $\frac{2}{1.41421356 - 1.41421356 i}$ by the conjugate of $1.41421356 - 1.41421356 i$ to make the denominator real.

$x = \frac{2}{1.41421356 - 1.41421356 i} \cdot \frac{1.41421356 + 1.41421356 i}{1.41421356 + 1.41421356 i}$

Multiply.

Combine.

$x = \frac{2 (1.41421356 + 1.41421356 i)}{(1.41421356 - 1.41421356 i) (1.41421356 + 1.41421356 i)}$

Simplify the numerator.

Apply the distributive property.

$x = \frac{2 \cdot 1.41421356 + 2 (1.41421356 i)}{(1.41421356 - 1.41421356 i) (1.41421356 + 1.41421356 i)}$

Multiply $2$ by $1.41421356$ .

$x = \frac{2.82842712 + 2 (1.41421356 i)}{(1.41421356 - 1.41421356 i) (1.41421356 + 1.41421356 i)}$

Multiply $1.41421356$ by $2$ .

$x = \frac{2.82842712 + 2.82842712 i}{(1.41421356 - 1.41421356 i) (1.41421356 + 1.41421356 i)}$

Simplify the denominator.

Expand $(1.41421356 - 1.41421356 i) (1.41421356 + 1.41421356 i)$ using the FOIL Method.

Apply the distributive property.

$x = \frac{2.82842712 + 2.82842712 i}{1.41421356 (1.41421356 + 1.41421356 i) - 1.41421356 i (1.41421356 + 1.41421356 i)}$

Apply the distributive property.

$x = \frac{2.82842712 + 2.82842712 i}{1.41421356 \cdot 1.41421356 + 1.41421356 (1.41421356 i) - 1.41421356 i (1.41421356 + 1.41421356 i)}$

Apply the distributive property.

$x = \frac{2.82842712 + 2.82842712 i}{1.41421356 \cdot 1.41421356 + 1.41421356 (1.41421356 i) - 1.41421356 i \cdot 1.41421356 - 1.41421356 i (1.41421356 i)}$

Simplify.

Multiply $1.41421356$ by $1.41421356$ .

$x = \frac{2.82842712 + 2.82842712 i}{2 + 1.41421356 (1.41421356 i) - 1.41421356 i \cdot 1.41421356 - 1.41421356 i (1.41421356 i)}$

Multiply $1.41421356$ by $1.41421356$ .

$x = \frac{2.82842712 + 2.82842712 i}{2 + 2 i - 1.41421356 i \cdot 1.41421356 - 1.41421356 i (1.41421356 i)}$

Multiply $1.41421356$ by $- 1.41421356$ .

$x = \frac{2.82842712 + 2.82842712 i}{2 + 2 i - 2 i - 1.41421356 i (1.41421356 i)}$

Multiply $1.41421356$ by $- 1.41421356$ .

$x = \frac{2.82842712 + 2.82842712 i}{2 + 2 i - 2 i - 2 i i}$

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

$x = \frac{2.82842712 + 2.82842712 i}{2 + 2 i - 2 i - 2 (i^{1} i)}$

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

$x = \frac{2.82842712 + 2.82842712 i}{2 + 2 i - 2 i - 2 (i^{1} i^{1})}$

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

$x = \frac{2.82842712 + 2.82842712 i}{2 + 2 i - 2 i - 2 i^{1 + 1}}$

Add $1$ and $1$ .

$x = \frac{2.82842712 + 2.82842712 i}{2 + 2 i - 2 i - 2 i^{2}}$

Subtract $2 i$ from $2 i$ .

$x = \frac{2.82842712 + 2.82842712 i}{2 + 0 i - 2 i^{2}}$

Simplify each term.

Multiply $0$ by $i$ .

$x = \frac{2.82842712 + 2.82842712 i}{2 + 0 - 2 i^{2}}$

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

$x = \frac{2.82842712 + 2.82842712 i}{2 + 0 - 2 \cdot - 1}$

Multiply $- 2$ by $- 1$ .

$x = \frac{2.82842712 + 2.82842712 i}{2 + 0 + 2}$

Add $2$ and $0$ .

$x = \frac{2.82842712 + 2.82842712 i}{2 + 2}$

Add $2$ and $2$ .

$x = \frac{2.82842712 + 2.82842712 i}{4}$

Rewrite $2.82842712$ as $1 (2.82842712)$ .

$x = \frac{1 (2.82842712) + 2.82842712 i}{4}$

Factor $1$ out of $2.82842712 i$ .

$x = \frac{1 (2.82842712) + 1 (2.82842712 i)}{4}$

Factor $1$ out of $1 (2.82842712) + 1 (2.82842712 i)$ .

$x = \frac{1 (2.82842712 + 2.82842712 i)}{4}$

Factor $4$ out of $4$ .

$x = \frac{1 (2.82842712 + 2.82842712 i)}{4 (1)}$

Separate fractions.

$x = \frac{1}{4} \cdot \frac{2.82842712 + 2.82842712 i}{1}$

Simplify the expression.

Divide $1$ by $4$ .

$x = 0.25 \frac{2.82842712 + 2.82842712 i}{1}$

Divide $2.82842712 + 2.82842712 i$ by $1$ .

$x = 0.25 (2.82842712 + 2.82842712 i)$

Apply the distributive property.

$x = 0.25 \cdot 2.82842712 + 0.25 (2.82842712 i)$

Multiply.

Multiply $0.25$ by $2.82842712$ .

$x = 0.70710678 + 0.25 (2.82842712 i)$

Multiply $2.82842712$ by $0.25$ .

$x = 0.70710678 + 0.70710678 i$

Solve for x n÷(( square root of 2)/2-( square root of 2i)/2)=nx

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