Algebra Examples

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

Step 1

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

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

Step 2

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

Tap for more steps...

Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

Tap for more steps...

Step 3.1

Simplify each term.

Tap for more steps...

Step 3.1.1

Combine using the product rule for radicals.

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

Step 3.1.2

Multiply $3$ by $3$ .

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

Step 3.1.3

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

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

Step 3.1.4

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

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

Step 3.1.5

Combine using the product rule for radicals.

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

Step 3.1.6

Multiply $3$ by $5$ .

$3 + \sqrt{15} + \sqrt{5} \sqrt{3} + \sqrt{5} \sqrt{5}$

Step 3.1.7

Combine using the product rule for radicals.

$3 + \sqrt{15} + \sqrt{5 \cdot 3} + \sqrt{5} \sqrt{5}$

Step 3.1.8

Multiply $5$ by $3$ .

$3 + \sqrt{15} + \sqrt{15} + \sqrt{5} \sqrt{5}$

Step 3.1.9

Combine using the product rule for radicals.

$3 + \sqrt{15} + \sqrt{15} + \sqrt{5 \cdot 5}$

Step 3.1.10

Multiply $5$ by $5$ .

$3 + \sqrt{15} + \sqrt{15} + \sqrt{25}$

Step 3.1.11

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

$3 + \sqrt{15} + \sqrt{15} + \sqrt{5^{2}}$

Step 3.1.12

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

$3 + \sqrt{15} + \sqrt{15} + 5$

Step 3.2

Add $3$ and $5$ .

$8 + \sqrt{15} + \sqrt{15}$

Step 3.3

Add $\sqrt{15}$ and $\sqrt{15}$ .

$8 + 2 \sqrt{15}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$8 + 2 \sqrt{15}$

Decimal Form:

$15.74596669 \dots$