Algebra Examples

$(\sqrt{6} + \sqrt{3}) (\sqrt{6} + \sqrt{3})$

Step 1

Expand $(\sqrt{6} + \sqrt{3}) (\sqrt{6} + \sqrt{3})$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

$\sqrt{6} (\sqrt{6} + \sqrt{3}) + \sqrt{3} (\sqrt{6} + \sqrt{3})$

Step 1.2

Apply the distributive property.

$\sqrt{6} \sqrt{6} + \sqrt{6} \sqrt{3} + \sqrt{3} (\sqrt{6} + \sqrt{3})$

Step 1.3

Apply the distributive property.

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

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

Combine using the product rule for radicals.

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

Step 2.1.2

Multiply $6$ by $6$ .

$\sqrt{36} + \sqrt{6} \sqrt{3} + \sqrt{3} \sqrt{6} + \sqrt{3} \sqrt{3}$

Step 2.1.3

Rewrite $36$ as $6^{2}$ .

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

Step 2.1.4

Pull terms out from under the radical.

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

Step 2.1.5

The absolute value is the distance between a number and zero. The distance between $0$ and $6$ is $6$ .

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

Step 2.1.6

Combine using the product rule for radicals.

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

Step 2.1.7

Multiply $6$ by $3$ .

$6 + \sqrt{18} + \sqrt{3} \sqrt{6} + \sqrt{3} \sqrt{3}$

Step 2.1.8

Rewrite $18$ as $3^{2} \cdot 2$ .

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Step 2.1.8.1

Factor $9$ out of $18$ .

$6 + \sqrt{9 (2)} + \sqrt{3} \sqrt{6} + \sqrt{3} \sqrt{3}$

Step 2.1.8.2

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

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

Step 2.1.9

Pull terms out from under the radical.

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

Step 2.1.10

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

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

Step 2.1.11

Combine using the product rule for radicals.

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

Step 2.1.12

Multiply $3$ by $6$ .

$6 + 3 \sqrt{2} + \sqrt{18} + \sqrt{3} \sqrt{3}$

Step 2.1.13

Rewrite $18$ as $3^{2} \cdot 2$ .

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Step 2.1.13.1

Factor $9$ out of $18$ .

$6 + 3 \sqrt{2} + \sqrt{9 (2)} + \sqrt{3} \sqrt{3}$

Step 2.1.13.2

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

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

Step 2.1.14

Pull terms out from under the radical.

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

Step 2.1.15

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

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

Step 2.1.16

Combine using the product rule for radicals.

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

Step 2.1.17

Multiply $3$ by $3$ .

$6 + 3 \sqrt{2} + 3 \sqrt{2} + \sqrt{9}$

Step 2.1.18

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

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

Step 2.1.19

Pull terms out from under the radical.

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

Step 2.1.20

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

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

Step 2.2

Add $6$ and $3$ .

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

Step 2.3

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

$9 + 6 \sqrt{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$9 + 6 \sqrt{2}$

Decimal Form:

$17.48528137 \dots$