Simplify the Radical Expression square root of 2( square root of 6+ square root of 10)

$\sqrt{2} (\sqrt{6} + \sqrt{10})$

Step 1

Apply the distributive property.

$\sqrt{2} \sqrt{6} + \sqrt{2} \sqrt{10}$

Step 2

Combine using the product rule for radicals.

$\sqrt{2 \cdot 6} + \sqrt{2} \sqrt{10}$

Step 3

Combine using the product rule for radicals.

$\sqrt{2 \cdot 6} + \sqrt{2 \cdot 10}$

Step 4

Simplify each term.

Multiply $2$ by $6$ .

$\sqrt{12} + \sqrt{2 \cdot 10}$

Rewrite $12$ as $2^{2} \cdot 3$ .

Factor $4$ out of $12$ .

$\sqrt{4 (3)} + \sqrt{2 \cdot 10}$

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

$\sqrt{2^{2} \cdot 3} + \sqrt{2 \cdot 10}$

Pull terms out from under the radical.

$| 2 | \sqrt{3} + \sqrt{2 \cdot 10}$

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$2 \sqrt{3} + \sqrt{2 \cdot 10}$

Multiply $2$ by $10$ .

$2 \sqrt{3} + \sqrt{20}$

Rewrite $20$ as $2^{2} \cdot 5$ .

Factor $4$ out of $20$ .

$2 \sqrt{3} + \sqrt{4 (5)}$

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

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

Pull terms out from under the radical.

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

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$7.93623757 \dots$