Algebra Examples

$(\sqrt{2} + i \sqrt{3}) (\sqrt{2} - i \sqrt{3})$

Step 1

Expand $(\sqrt{2} + i \sqrt{3}) (\sqrt{2} - i \sqrt{3})$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

$\sqrt{2} (\sqrt{2} - i \sqrt{3}) + i \sqrt{3} (\sqrt{2} - i \sqrt{3})$

Step 1.2

Apply the distributive property.

$\sqrt{2} \sqrt{2} + \sqrt{2} (- i \sqrt{3}) + i \sqrt{3} (\sqrt{2} - i \sqrt{3})$

Step 1.3

Apply the distributive property.

$\sqrt{2} \sqrt{2} + \sqrt{2} (- i \sqrt{3}) + i \sqrt{3} \sqrt{2} + i \sqrt{3} (- i \sqrt{3})$

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.

$\sqrt{2 \cdot 2} + \sqrt{2} (- i \sqrt{3}) + i \sqrt{3} \sqrt{2} + i \sqrt{3} (- i \sqrt{3})$

Step 2.1.2

Multiply $2$ by $2$ .

$\sqrt{4} + \sqrt{2} (- i \sqrt{3}) + i \sqrt{3} \sqrt{2} + i \sqrt{3} (- i \sqrt{3})$

Step 2.1.3

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

$\sqrt{2^{2}} + \sqrt{2} (- i \sqrt{3}) + i \sqrt{3} \sqrt{2} + i \sqrt{3} (- i \sqrt{3})$

Step 2.1.4

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

$2 + \sqrt{2} (- i \sqrt{3}) + i \sqrt{3} \sqrt{2} + i \sqrt{3} (- i \sqrt{3})$

Step 2.1.5

Multiply $\sqrt{2} (- i \sqrt{3})$ .

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Step 2.1.5.1

Combine using the product rule for radicals.

$2 - i \sqrt{2 \cdot 3} + i \sqrt{3} \sqrt{2} + i \sqrt{3} (- i \sqrt{3})$

Step 2.1.5.2

Multiply $2$ by $3$ .

$2 - i \sqrt{6} + i \sqrt{3} \sqrt{2} + i \sqrt{3} (- i \sqrt{3})$

Step 2.1.6

Multiply $i \sqrt{3} \sqrt{2}$ .

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Step 2.1.6.1

Combine using the product rule for radicals.

$2 - i \sqrt{6} + i \sqrt{2 \cdot 3} + i \sqrt{3} (- i \sqrt{3})$

Step 2.1.6.2

Multiply $2$ by $3$ .

$2 - i \sqrt{6} + i \sqrt{6} + i \sqrt{3} (- i \sqrt{3})$

Step 2.1.7

Multiply $i \sqrt{3} (- i \sqrt{3})$ .

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Step 2.1.7.1

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

$2 - i \sqrt{6} + i \sqrt{6} + \sqrt{3} (- (i^{1} i) \sqrt{3})$

Step 2.1.7.2

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

$2 - i \sqrt{6} + i \sqrt{6} + \sqrt{3} (- (i^{1} i^{1}) \sqrt{3})$

Step 2.1.7.3

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

$2 - i \sqrt{6} + i \sqrt{6} + \sqrt{3} (- i^{1 + 1} \sqrt{3})$

Step 2.1.7.4

Add $1$ and $1$ .

$2 - i \sqrt{6} + i \sqrt{6} + \sqrt{3} (- i^{2} \sqrt{3})$

Step 2.1.7.5

Raise $\sqrt{3}$ to the power of $1$ .

$2 - i \sqrt{6} + i \sqrt{6} - i^{2} ({\sqrt{3}}^{1} \sqrt{3})$

Step 2.1.7.6

Raise $\sqrt{3}$ to the power of $1$ .

$2 - i \sqrt{6} + i \sqrt{6} - i^{2} ({\sqrt{3}}^{1} {\sqrt{3}}^{1})$

Step 2.1.7.7

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

$2 - i \sqrt{6} + i \sqrt{6} - i^{2} {\sqrt{3}}^{1 + 1}$

Step 2.1.7.8

Add $1$ and $1$ .

$2 - i \sqrt{6} + i \sqrt{6} - i^{2} {\sqrt{3}}^{2}$

Step 2.1.8

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

$2 - i \sqrt{6} + i \sqrt{6} - - 1 {\sqrt{3}}^{2}$

Step 2.1.9

Multiply $- 1$ by $- 1$ .

$2 - i \sqrt{6} + i \sqrt{6} + 1 {\sqrt{3}}^{2}$

Step 2.1.10

Multiply ${\sqrt{3}}^{2}$ by $1$ .

$2 - i \sqrt{6} + i \sqrt{6} + {\sqrt{3}}^{2}$

Step 2.1.11

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 2.1.11.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$2 - i \sqrt{6} + i \sqrt{6} + {(3^{\frac{1}{2}})}^{2}$

Step 2.1.11.2

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

$2 - i \sqrt{6} + i \sqrt{6} + 3^{\frac{1}{2} \cdot 2}$

Step 2.1.11.3

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

$2 - i \sqrt{6} + i \sqrt{6} + 3^{\frac{2}{2}}$

Step 2.1.11.4

Cancel the common factor of $2$ .

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Step 2.1.11.4.1

Cancel the common factor.

$2 - i \sqrt{6} + i \sqrt{6} + 3^{\frac{2}{2}}$

Step 2.1.11.4.2

Rewrite the expression.

$2 - i \sqrt{6} + i \sqrt{6} + 3^{1}$

Step 2.1.11.5

Evaluate the exponent.

$2 - i \sqrt{6} + i \sqrt{6} + 3$

Step 2.2

Add $2$ and $3$ .

$- i \sqrt{6} + i \sqrt{6} + 5$

Step 2.3

Add $- i \sqrt{6}$ and $i \sqrt{6}$ .

$0 + 5$

Step 2.4

Add $0$ and $5$ .

$5$