Algebra Examples

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

Step 1

Expand $(\frac{\sqrt{2}}{2} + i \sqrt{2}) (\frac{\sqrt{2}}{2} - i \sqrt{2})$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

Step 2

Simplify and combine like terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Multiply $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2}}{2}$ .

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Step 2.1.1.1

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

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

Step 2.1.1.2

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

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

Step 2.1.1.3

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

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

Step 2.1.1.4

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

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

Step 2.1.1.5

Add $1$ and $1$ .

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

Step 2.1.1.6

Multiply $2$ by $2$ .

$\frac{{\sqrt{2}}^{2}}{4} + \frac{\sqrt{2}}{2} (- i \sqrt{2}) + i \sqrt{2} \frac{\sqrt{2}}{2} + i \sqrt{2} (- i \sqrt{2})$

Step 2.1.2

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

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Step 2.1.2.1

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

$\frac{{(2^{\frac{1}{2}})}^{2}}{4} + \frac{\sqrt{2}}{2} (- i \sqrt{2}) + i \sqrt{2} \frac{\sqrt{2}}{2} + i \sqrt{2} (- i \sqrt{2})$

Step 2.1.2.2

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

$\frac{2^{\frac{1}{2} \cdot 2}}{4} + \frac{\sqrt{2}}{2} (- i \sqrt{2}) + i \sqrt{2} \frac{\sqrt{2}}{2} + i \sqrt{2} (- i \sqrt{2})$

Step 2.1.2.3

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

$\frac{2^{\frac{2}{2}}}{4} + \frac{\sqrt{2}}{2} (- i \sqrt{2}) + i \sqrt{2} \frac{\sqrt{2}}{2} + i \sqrt{2} (- i \sqrt{2})$

Step 2.1.2.4

Cancel the common factor of $2$ .

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Step 2.1.2.4.1

Cancel the common factor.

$\frac{2^{\frac{2}{2}}}{4} + \frac{\sqrt{2}}{2} (- i \sqrt{2}) + i \sqrt{2} \frac{\sqrt{2}}{2} + i \sqrt{2} (- i \sqrt{2})$

Step 2.1.2.4.2

Rewrite the expression.

$\frac{2^{1}}{4} + \frac{\sqrt{2}}{2} (- i \sqrt{2}) + i \sqrt{2} \frac{\sqrt{2}}{2} + i \sqrt{2} (- i \sqrt{2})$

Step 2.1.2.5

Evaluate the exponent.

$\frac{2}{4} + \frac{\sqrt{2}}{2} (- i \sqrt{2}) + i \sqrt{2} \frac{\sqrt{2}}{2} + i \sqrt{2} (- i \sqrt{2})$

Step 2.1.3

Cancel the common factor of $2$ and $4$ .

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Step 2.1.3.1

Factor $2$ out of $2$ .

$\frac{2 (1)}{4} + \frac{\sqrt{2}}{2} (- i \sqrt{2}) + i \sqrt{2} \frac{\sqrt{2}}{2} + i \sqrt{2} (- i \sqrt{2})$

Step 2.1.3.2

Cancel the common factors.

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Step 2.1.3.2.1

Factor $2$ out of $4$ .

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

Step 2.1.3.2.2

Cancel the common factor.

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

Step 2.1.3.2.3

Rewrite the expression.

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

Step 2.1.4

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

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Step 2.1.4.1

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

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

Step 2.1.4.2

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

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

Step 2.1.4.3

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

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

Step 2.1.4.4

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

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

Step 2.1.4.5

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

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

Step 2.1.4.6

Add $1$ and $1$ .

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

Step 2.1.5

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

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Step 2.1.5.1

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

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

Step 2.1.5.2

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

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

Step 2.1.5.3

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

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

Step 2.1.5.4

Cancel the common factor of $2$ .

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Step 2.1.5.4.1

Cancel the common factor.

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

Step 2.1.5.4.2

Rewrite the expression.

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

Step 2.1.5.5

Evaluate the exponent.

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

Step 2.1.6

Cancel the common factor of $2$ .

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Step 2.1.6.1

Cancel the common factor.

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

Step 2.1.6.2

Divide $i$ by $1$ .

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

Step 2.1.7

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

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Step 2.1.7.1

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

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

Step 2.1.7.2

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

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

Step 2.1.7.3

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

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

Step 2.1.7.4

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

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

Step 2.1.7.5

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

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

Step 2.1.7.6

Add $1$ and $1$ .

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

Step 2.1.8

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

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Step 2.1.8.1

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

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

Step 2.1.8.2

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

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

Step 2.1.8.3

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

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

Step 2.1.8.4

Cancel the common factor of $2$ .

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Step 2.1.8.4.1

Cancel the common factor.

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

Step 2.1.8.4.2

Rewrite the expression.

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

Step 2.1.8.5

Evaluate the exponent.

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

Step 2.1.9

Cancel the common factor of $2$ .

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Step 2.1.9.1

Cancel the common factor.

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

Step 2.1.9.2

Divide $i$ by $1$ .

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

Step 2.1.10

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

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Step 2.1.10.1

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

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

Step 2.1.10.2

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

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

Step 2.1.10.3

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

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

Step 2.1.10.4

Add $1$ and $1$ .

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

Step 2.1.10.5

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

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

Step 2.1.10.6

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

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

Step 2.1.10.7

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

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

Step 2.1.10.8

Add $1$ and $1$ .

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

Step 2.1.11

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

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

Step 2.1.12

Multiply $- 1$ by $- 1$ .

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

Step 2.1.13

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

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

Step 2.1.14

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

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Step 2.1.14.1

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

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

Step 2.1.14.2

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

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

Step 2.1.14.3

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

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

Step 2.1.14.4

Cancel the common factor of $2$ .

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Step 2.1.14.4.1

Cancel the common factor.

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

Step 2.1.14.4.2

Rewrite the expression.

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

Step 2.1.14.5

Evaluate the exponent.

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 2.5

Simplify the numerator.

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Step 2.5.1

Multiply $2$ by $2$ .

$\frac{1 + 4}{2} - i + i$

Step 2.5.2

Add $1$ and $4$ .

$\frac{5}{2} - i + i$

Step 2.6

Add $- i$ and $i$ .

$\frac{5}{2} + 0$

Step 2.7

Add $\frac{5}{2}$ and $0$ .

$\frac{5}{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{5}{2}$

Decimal Form:

$2.5$

Mixed Number Form:

$2 \frac{1}{2}$