Algebra Examples

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

Step 1

Rewrite ${(\sqrt{2} + \sqrt{3})}^{2}$ as $(\sqrt{2} + \sqrt{3}) (\sqrt{2} + \sqrt{3})$ .

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

Step 2

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

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Combine using the product rule for radicals.

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

Step 3.1.2

Multiply $2$ by $2$ .

$\sqrt{4} + \sqrt{2} \sqrt{3} + \sqrt{3} \sqrt{2} + \sqrt{3} \sqrt{3}$

Step 3.1.3

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

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

Step 3.1.4

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

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

Step 3.1.5

Combine using the product rule for radicals.

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

Step 3.1.6

Multiply $2$ by $3$ .

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

Step 3.1.7

Combine using the product rule for radicals.

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

Step 3.1.8

Multiply $3$ by $2$ .

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

Step 3.1.9

Combine using the product rule for radicals.

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

Step 3.1.10

Multiply $3$ by $3$ .

$2 + \sqrt{6} + \sqrt{6} + \sqrt{9}$

Step 3.1.11

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

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

Step 3.1.12

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

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

Step 3.2

Add $2$ and $3$ .

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

Step 3.3

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

$5 + 2 \sqrt{6}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$5 + 2 \sqrt{6}$

Decimal Form:

$9.89897948 \dots$