Evaluate (8+ square root of 2)^2

${(8 + \sqrt{2})}^{2}$

Step 1

Rewrite ${(8 + \sqrt{2})}^{2}$ as $(8 + \sqrt{2}) (8 + \sqrt{2})$ .

$(8 + \sqrt{2}) (8 + \sqrt{2})$

Step 2

Expand $(8 + \sqrt{2}) (8 + \sqrt{2})$ using the FOIL Method.

Apply the distributive property.

$8 (8 + \sqrt{2}) + \sqrt{2} (8 + \sqrt{2})$

Apply the distributive property.

$8 \cdot 8 + 8 \sqrt{2} + \sqrt{2} (8 + \sqrt{2})$

Apply the distributive property.

$8 \cdot 8 + 8 \sqrt{2} + \sqrt{2} \cdot 8 + \sqrt{2} \sqrt{2}$

Step 3

Simplify and combine like terms.

Simplify each term.

Multiply $8$ by $8$ .

$64 + 8 \sqrt{2} + \sqrt{2} \cdot 8 + \sqrt{2} \sqrt{2}$

Move $8$ to the left of $\sqrt{2}$ .

$64 + 8 \sqrt{2} + 8 \cdot \sqrt{2} + \sqrt{2} \sqrt{2}$

Combine using the product rule for radicals.

$64 + 8 \sqrt{2} + 8 \sqrt{2} + \sqrt{2 \cdot 2}$

Multiply $2$ by $2$ .

$64 + 8 \sqrt{2} + 8 \sqrt{2} + \sqrt{4}$

Rewrite $4$ as $2^{2}$ .

$64 + 8 \sqrt{2} + 8 \sqrt{2} + \sqrt{2^{2}}$

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

$64 + 8 \sqrt{2} + 8 \sqrt{2} + 2$

Add $64$ and $2$ .

$66 + 8 \sqrt{2} + 8 \sqrt{2}$

Add $8 \sqrt{2}$ and $8 \sqrt{2}$ .

$66 + 16 \sqrt{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$66 + 16 \sqrt{2}$

Decimal Form:

$88.62741699 \dots$