Simplify the Radical Expression (3+ square root of 6)^2

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

Rewrite ${(3 + \sqrt{6})}^{2}$ as $(3 + \sqrt{6}) (3 + \sqrt{6})$ .

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $3$ by $3$ .

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

Move $3$ to the left of $\sqrt{6}$ .

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

Combine using the product rule for radicals.

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

Multiply $6$ by $6$ .

$9 + 3 \sqrt{6} + 3 \sqrt{6} + \sqrt{36}$

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

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

Pull terms out from under the radical.

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

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

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

Add $9$ and $6$ .

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

Add $3 \sqrt{6}$ and $3 \sqrt{6}$ .

$15 + 6 \sqrt{6}$

The result can be shown in multiple forms.

Exact Form:

$15 + 6 \sqrt{6}$

Decimal Form:

$29.69693845 \dots$