Evaluate (7+ square root of 7)^2

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

Step 1

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

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

Step 2

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

Simplify each term.

Multiply $7$ by $7$ .

$49 + 7 \sqrt{7} + \sqrt{7} \cdot 7 + \sqrt{7} \sqrt{7}$

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

$49 + 7 \sqrt{7} + 7 \cdot \sqrt{7} + \sqrt{7} \sqrt{7}$

Combine using the product rule for radicals.

$49 + 7 \sqrt{7} + 7 \sqrt{7} + \sqrt{7 \cdot 7}$

Multiply $7$ by $7$ .

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

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

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

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

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

Add $49$ and $7$ .

$56 + 7 \sqrt{7} + 7 \sqrt{7}$

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

$56 + 14 \sqrt{7}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$56 + 14 \sqrt{7}$

Decimal Form:

$93.04051835 \dots$