Algebra Examples

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

Step 1

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

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Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

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

Combine using the product rule for radicals.

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

Step 2.1.2

Multiply $2$ by $2$ .

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

Step 2.1.3

Rewrite $4$ as $2^{2}$ .

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

Step 2.1.4

Pull terms out from under the radical, assuming positive real numbers.

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

Step 2.1.5

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

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Step 2.1.5.1

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

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

Step 2.1.5.2

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

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

Step 2.1.5.3

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

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

Step 2.1.5.4

Add $1$ and $1$ .

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

Step 2.1.6

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

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Step 2.1.6.1

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

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

Step 2.1.6.2

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

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

Step 2.1.6.3

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

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

Step 2.1.6.4

Cancel the common factor of $2$ .

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Step 2.1.6.4.1

Cancel the common factor.

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

Step 2.1.6.4.2

Rewrite the expression.

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

Step 2.1.6.5

Evaluate the exponent.

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

Step 2.1.7

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

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Step 2.1.7.1

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

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

Step 2.1.7.2

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

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

Step 2.1.7.3

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

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

Step 2.1.7.4

Add $1$ and $1$ .

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

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 $2^{\frac{1}{2}}$ .

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

Step 2.1.8.2

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

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

Step 2.1.8.3

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

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

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.

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

Step 2.1.8.4.2

Rewrite the expression.

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

Step 2.1.8.5

Evaluate the exponent.

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

Step 2.1.9

Multiply $2$ by $- 1$ .

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

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 $1$ .

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

Step 2.1.10.2

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

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

Step 2.1.10.3

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

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

Step 2.1.10.4

Add $1$ and $1$ .

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

Step 2.1.10.5

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

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

Step 2.1.10.6

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

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

Step 2.1.10.7

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

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

Step 2.1.10.8

Add $1$ and $1$ .

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

Step 2.1.11

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

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

Step 2.1.12

Multiply $- 1$ by $- 1$ .

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

Step 2.1.13

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

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

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 $2^{\frac{1}{2}}$ .

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

Step 2.1.14.2

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

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

Step 2.1.14.3

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

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

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.

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

Step 2.1.14.4.2

Rewrite the expression.

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

Step 2.1.14.5

Evaluate the exponent.

$2 + 2 i - 2 i + 2$

Step 2.2

Add $2$ and $2$ .

$4 + 2 i - 2 i$

Step 2.3

Subtract $2 i$ from $2 i$ .

$4 + 0$

Step 2.4

Add $4$ and $0$ .

$4$