Algebra Examples

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

Step 1

Expand $(\sqrt{3} + \sqrt{2}) (\sqrt{3} + \sqrt{5})$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

Step 2

Simplify each term.

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Step 2.1

Combine using the product rule for radicals.

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

Step 2.2

Multiply $3$ by $3$ .

$\sqrt{9} + \sqrt{3} \sqrt{5} + \sqrt{2} \sqrt{3} + \sqrt{2} \sqrt{5}$

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Combine using the product rule for radicals.

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

Step 2.6

Multiply $3$ by $5$ .

$3 + \sqrt{15} + \sqrt{2} \sqrt{3} + \sqrt{2} \sqrt{5}$

Step 2.7

Combine using the product rule for radicals.

$3 + \sqrt{15} + \sqrt{2 \cdot 3} + \sqrt{2} \sqrt{5}$

Step 2.8

Multiply $2$ by $3$ .

$3 + \sqrt{15} + \sqrt{6} + \sqrt{2} \sqrt{5}$

Step 2.9

Combine using the product rule for radicals.

$3 + \sqrt{15} + \sqrt{6} + \sqrt{2 \cdot 5}$

Step 2.10

Multiply $2$ by $5$ .

$3 + \sqrt{15} + \sqrt{6} + \sqrt{10}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$3 + \sqrt{15} + \sqrt{6} + \sqrt{10}$

Decimal Form:

$12.48475074 \dots$