Simplify (5+ square root of 3)^2

${(5 + \sqrt{3})}^{2}$

Step 1

Rewrite ${(5 + \sqrt{3})}^{2}$ as $(5 + \sqrt{3}) (5 + \sqrt{3})$ .

$(5 + \sqrt{3}) (5 + \sqrt{3})$

Step 2

Expand $(5 + \sqrt{3}) (5 + \sqrt{3})$ using the FOIL Method.

Apply the distributive property.

$5 (5 + \sqrt{3}) + \sqrt{3} (5 + \sqrt{3})$

Apply the distributive property.

$5 \cdot 5 + 5 \sqrt{3} + \sqrt{3} (5 + \sqrt{3})$

Apply the distributive property.

$5 \cdot 5 + 5 \sqrt{3} + \sqrt{3} \cdot 5 + \sqrt{3} \sqrt{3}$

Step 3

Simplify and combine like terms.

Simplify each term.

Multiply $5$ by $5$ .

$25 + 5 \sqrt{3} + \sqrt{3} \cdot 5 + \sqrt{3} \sqrt{3}$

Move $5$ to the left of $\sqrt{3}$ .

$25 + 5 \sqrt{3} + 5 \cdot \sqrt{3} + \sqrt{3} \sqrt{3}$

Combine using the product rule for radicals.

$25 + 5 \sqrt{3} + 5 \sqrt{3} + \sqrt{3 \cdot 3}$

Multiply $3$ by $3$ .

$25 + 5 \sqrt{3} + 5 \sqrt{3} + \sqrt{9}$

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

$25 + 5 \sqrt{3} + 5 \sqrt{3} + \sqrt{3^{2}}$

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

$25 + 5 \sqrt{3} + 5 \sqrt{3} + 3$

Add $25$ and $3$ .

$28 + 5 \sqrt{3} + 5 \sqrt{3}$

Add $5 \sqrt{3}$ and $5 \sqrt{3}$ .

$28 + 10 \sqrt{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$28 + 10 \sqrt{3}$

Decimal Form:

$45.32050807 \dots$