Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

Simplify each term.

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Step 1.1.1

Rewrite $\sqrt{\frac{1}{2}}$ as $\frac{\sqrt{1}}{\sqrt{2}}$ .

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

Step 1.1.2

Any root of $1$ is $1$ .

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

Step 1.1.3

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

$\frac{1 + i \sqrt{2}}{2} - (1 - i (\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}))$

Step 1.1.4

Combine and simplify the denominator.

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Step 1.1.4.1

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

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

Step 1.1.4.2

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 1.1.4.3

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 1.1.4.4

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

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

Step 1.1.4.5

Add $1$ and $1$ .

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

Step 1.1.4.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 1.1.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

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

Step 1.1.4.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 1.1.4.6.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 1.1.4.6.4

Cancel the common factor of $2$ .

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Step 1.1.4.6.4.1

Cancel the common factor.

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

Step 1.1.4.6.4.2

Rewrite the expression.

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

Step 1.1.4.6.5

Evaluate the exponent.

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

Step 1.1.5

Combine $\frac{\sqrt{2}}{2}$ and $i$ .

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Multiply $- 1$ by $1$ .

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

Step 1.4

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

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Step 1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 1.4.2

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

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

Step 2

Simplify terms.

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Step 2.1

Combine the numerators over the common denominator.

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

Step 2.2

Reorder the factors of $\sqrt{2} i$ .

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

Step 2.3

Add $i \sqrt{2}$ and $i \sqrt{2}$ .

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

Step 3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 4

Combine $- 1$ and $\frac{2}{2}$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Move the negative in front of the fraction.

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