Evaluate (4+ square root of 7)^2

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

Step 1

Rewrite ${(4 + \sqrt{7})}^{2}$ as $(4 + \sqrt{7}) (4 + \sqrt{7})$ .

$(4 + \sqrt{7}) (4 + \sqrt{7})$

Step 2

Expand $(4 + \sqrt{7}) (4 + \sqrt{7})$ using the FOIL Method.

Apply the distributive property.

$4 (4 + \sqrt{7}) + \sqrt{7} (4 + \sqrt{7})$

Apply the distributive property.

$4 \cdot 4 + 4 \sqrt{7} + \sqrt{7} (4 + \sqrt{7})$

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

Simplify each term.

Multiply $4$ by $4$ .

$16 + 4 \sqrt{7} + \sqrt{7} \cdot 4 + \sqrt{7} \sqrt{7}$

Move $4$ to the left of $\sqrt{7}$ .

$16 + 4 \sqrt{7} + 4 \cdot \sqrt{7} + \sqrt{7} \sqrt{7}$

Combine using the product rule for radicals.

$16 + 4 \sqrt{7} + 4 \sqrt{7} + \sqrt{7 \cdot 7}$

Multiply $7$ by $7$ .

$16 + 4 \sqrt{7} + 4 \sqrt{7} + \sqrt{49}$

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

$16 + 4 \sqrt{7} + 4 \sqrt{7} + \sqrt{7^{2}}$

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

$16 + 4 \sqrt{7} + 4 \sqrt{7} + 7$

Add $16$ and $7$ .

$23 + 4 \sqrt{7} + 4 \sqrt{7}$

Add $4 \sqrt{7}$ and $4 \sqrt{7}$ .

$23 + 8 \sqrt{7}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$23 + 8 \sqrt{7}$

Decimal Form:

$44.16601048 \dots$