Algebra Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

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.

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

Step 3.1.2

Multiply $3$ by $3$ .

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

Step 3.1.3

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

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

Step 3.1.4

Pull terms out from under the radical.

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

Step 3.1.5

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

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

Step 3.1.6

Combine using the product rule for radicals.

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

Step 3.1.7

Multiply $3$ by $2$ .

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

Step 3.1.8

Combine using the product rule for radicals.

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

Step 3.1.9

Multiply $2$ by $3$ .

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

Step 3.1.10

Combine using the product rule for radicals.

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

Step 3.1.11

Multiply $2$ by $2$ .

$3 + \sqrt{6} + \sqrt{6} + \sqrt{4}$

Step 3.1.12

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

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

Step 3.1.13

Pull terms out from under the radical.

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

Step 3.1.14

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

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

Step 3.2

Add $3$ and $2$ .

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

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:

$9.89897948 \dots$