Finite Math Examples

$(2 + \sqrt{3}) (1 + \sqrt{3})$

Step 1

Expand $(2 + \sqrt{3}) (1 + \sqrt{3})$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

$2 (1 + \sqrt{3}) + \sqrt{3} (1 + \sqrt{3})$

Step 1.2

Apply the distributive property.

$2 \cdot 1 + 2 \sqrt{3} + \sqrt{3} (1 + \sqrt{3})$

Step 1.3

Apply the distributive property.

$2 \cdot 1 + 2 \sqrt{3} + \sqrt{3} \cdot 1 + \sqrt{3} \sqrt{3}$

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

Multiply $2$ by $1$ .

$2 + 2 \sqrt{3} + \sqrt{3} \cdot 1 + \sqrt{3} \sqrt{3}$

Step 2.1.2

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

$2 + 2 \sqrt{3} + \sqrt{3} + \sqrt{3} \sqrt{3}$

Step 2.1.3

Combine using the product rule for radicals.

$2 + 2 \sqrt{3} + \sqrt{3} + \sqrt{3 \cdot 3}$

Step 2.1.4

Multiply $3$ by $3$ .

$2 + 2 \sqrt{3} + \sqrt{3} + \sqrt{9}$

Step 2.1.5

Rewrite $9$ as $3^{2}$ .

$2 + 2 \sqrt{3} + \sqrt{3} + \sqrt{3^{2}}$

Step 2.1.6

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

$2 + 2 \sqrt{3} + \sqrt{3} + 3$

Step 2.2

Add $2$ and $3$ .

$5 + 2 \sqrt{3} + \sqrt{3}$

Step 2.3

Add $2 \sqrt{3}$ and $\sqrt{3}$ .

$5 + 3 \sqrt{3}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$5 + 3 \sqrt{3}$

Decimal Form:

$10.19615242 \dots$