Algebra Examples

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

Step 1

To find the reciprocal, divide $1$ by the number given.

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

Step 2

Simplify.

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Step 2.1

Combine the numerators over the common denominator.

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

Step 2.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.3

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

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

Step 2.4

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.

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

Step 2.5

Multiply.

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Step 2.5.1

Combine.

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

Step 2.5.2

Simplify the numerator.

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Step 2.5.2.1

Apply the distributive property.

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

Step 2.5.2.2

Multiply $2$ by $1.41421356$ .

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

Step 2.5.2.3

Multiply $1.41421356$ by $2$ .

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

Step 2.5.3

Simplify the denominator.

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Step 2.5.3.1

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

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Step 2.5.3.1.1

Apply the distributive property.

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

Step 2.5.3.1.2

Apply the distributive property.

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

Step 2.5.3.1.3

Apply the distributive property.

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

Step 2.5.3.2

Simplify.

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Step 2.5.3.2.1

Multiply $1.41421356$ by $1.41421356$ .

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

Step 2.5.3.2.2

Multiply $1.41421356$ by $1.41421356$ .

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

Step 2.5.3.2.3

Multiply $1.41421356$ by $- 1.41421356$ .

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

Step 2.5.3.2.4

Multiply $1.41421356$ by $- 1.41421356$ .

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

Step 2.5.3.2.5

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

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

Step 2.5.3.2.6

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

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

Step 2.5.3.2.7

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

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

Step 2.5.3.2.8

Add $1$ and $1$ .

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

Step 2.5.3.2.9

Subtract $2 i$ from $2 i$ .

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

Step 2.5.3.3

Simplify each term.

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Step 2.5.3.3.1

Multiply $0$ by $i$ .

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

Step 2.5.3.3.2

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

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

Step 2.5.3.3.3

Multiply $- 2$ by $- 1$ .

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

Step 2.5.3.4

Add $2$ and $0$ .

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

Step 2.5.3.5

Add $2$ and $2$ .

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

Step 2.6

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

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

Step 2.7

Factor $1$ out of $2.82842712 i$ .

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

Step 2.8

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

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

Step 2.9

Factor $4$ out of $4$ .

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

Step 2.10

Separate fractions.

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

Step 2.11

Simplify the expression.

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Step 2.11.1

Divide $1$ by $4$ .

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

Step 2.11.2

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

$0.25 (2.82842712 + 2.82842712 i)$

Step 2.12

Apply the distributive property.

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

Step 2.13

Multiply.

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Step 2.13.1

Multiply $0.25$ by $2.82842712$ .

$0.70710678 + 0.25 (2.82842712 i)$

Step 2.13.2

Multiply $2.82842712$ by $0.25$ .

$0.70710678 + 0.70710678 i$