Precalculus Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $3$ by $3$ .

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

Step 3.1.2

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

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

Step 3.1.3

Combine using the product rule for radicals.

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

Step 3.1.4

Multiply $5$ by $5$ .

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

Step 3.1.5

Rewrite $25$ as $5^{2}$ .

$9 + 3 \sqrt{5} + 3 \sqrt{5} + \sqrt{5^{2}}$

Step 3.1.6

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

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

Step 3.2

Add $9$ and $5$ .

$14 + 3 \sqrt{5} + 3 \sqrt{5}$

Step 3.3

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

$14 + 6 \sqrt{5}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$14 + 6 \sqrt{5}$

Decimal Form:

$27.41640786 \dots$