Evaluate (2+ square root of 7)^2

${(2 + \sqrt{7})}^{2}$

Step 1

Rewrite ${(2 + \sqrt{7})}^{2}$ as $(2 + \sqrt{7}) (2 + \sqrt{7})$ .

$(2 + \sqrt{7}) (2 + \sqrt{7})$

Step 2

Expand $(2 + \sqrt{7}) (2 + \sqrt{7})$ using the FOIL Method.

Apply the distributive property.

$2 (2 + \sqrt{7}) + \sqrt{7} (2 + \sqrt{7})$

Apply the distributive property.

$2 \cdot 2 + 2 \sqrt{7} + \sqrt{7} (2 + \sqrt{7})$

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

Simplify each term.

Multiply $2$ by $2$ .

$4 + 2 \sqrt{7} + \sqrt{7} \cdot 2 + \sqrt{7} \sqrt{7}$

Move $2$ to the left of $\sqrt{7}$ .

$4 + 2 \sqrt{7} + 2 \cdot \sqrt{7} + \sqrt{7} \sqrt{7}$

Combine using the product rule for radicals.

$4 + 2 \sqrt{7} + 2 \sqrt{7} + \sqrt{7 \cdot 7}$

Multiply $7$ by $7$ .

$4 + 2 \sqrt{7} + 2 \sqrt{7} + \sqrt{49}$

Rewrite $49$ as $7^{2}$ .

$4 + 2 \sqrt{7} + 2 \sqrt{7} + \sqrt{7^{2}}$

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

$4 + 2 \sqrt{7} + 2 \sqrt{7} + 7$

Add $4$ and $7$ .

$11 + 2 \sqrt{7} + 2 \sqrt{7}$

Add $2 \sqrt{7}$ and $2 \sqrt{7}$ .

$11 + 4 \sqrt{7}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$11 + 4 \sqrt{7}$

Decimal Form:

$21.58300524 \dots$