Finite Math Examples

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

Step 1

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

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Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

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.

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

Step 2.1.2

Multiply $2$ by $2$ .

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

Step 2.1.3

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

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

Step 2.1.4

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

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

Step 2.1.5

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

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Step 2.1.5.1

Combine using the product rule for radicals.

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

Step 2.1.5.2

Multiply $2$ by $3$ .

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

Step 2.1.6

Combine using the product rule for radicals.

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

Step 2.1.7

Multiply $2$ by $3$ .

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

Step 2.1.8

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

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Step 2.1.8.1

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

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

Step 2.1.8.2

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

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

Step 2.1.8.3

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

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

Step 2.1.8.4

Add $1$ and $1$ .

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

Step 2.1.9

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

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Step 2.1.9.1

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

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

Step 2.1.9.2

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

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

Step 2.1.9.3

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

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

Step 2.1.9.4

Cancel the common factor of $2$ .

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Step 2.1.9.4.1

Cancel the common factor.

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

Step 2.1.9.4.2

Rewrite the expression.

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

Step 2.1.9.5

Simplify.

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

Step 2.1.10

Multiply $3$ by $- 1$ .

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

Step 2.2

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

$2 + 0 - 3 i$

Step 2.3

Add $2$ and $0$ .

$2 - 3 i$