Simplify (2+ square root of 3)^2

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

Step 1

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

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

Step 2

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

Simplify each term.

Multiply $2$ by $2$ .

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

Move $2$ to the left of $\sqrt{3}$ .

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

Combine using the product rule for radicals.

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

Multiply $3$ by $3$ .

$4 + 2 \sqrt{3} + 2 \sqrt{3} + \sqrt{9}$

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

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

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

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

Add $4$ and $3$ .

$7 + 2 \sqrt{3} + 2 \sqrt{3}$

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

$7 + 4 \sqrt{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$7 + 4 \sqrt{3}$

Decimal Form:

$13.92820323 \dots$