Simplify the Radical Expression - square root of 2( square root of 10-4 square root of 6)

$- \sqrt{2} (\sqrt{10} - 4 \sqrt{6})$

Step 1

Apply the distributive property.

$- \sqrt{2} \sqrt{10} - \sqrt{2} (- 4 \sqrt{6})$

Step 2

Multiply $- \sqrt{2} \sqrt{10}$ .

Combine using the product rule for radicals.

$- \sqrt{10 \cdot 2} - \sqrt{2} (- 4 \sqrt{6})$

Multiply $10$ by $2$ .

$- \sqrt{20} - \sqrt{2} (- 4 \sqrt{6})$

Step 3

Multiply $- \sqrt{2} (- 4 \sqrt{6})$ .

Multiply $- 4$ by $- 1$ .

$- \sqrt{20} + 4 \sqrt{2} \sqrt{6}$

Combine using the product rule for radicals.

$- \sqrt{20} + 4 \sqrt{6 \cdot 2}$

Multiply $6$ by $2$ .

$- \sqrt{20} + 4 \sqrt{12}$

Step 4

Simplify each term.

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

Factor $4$ out of $20$ .

$- \sqrt{4 (5)} + 4 \sqrt{12}$

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

$- \sqrt{2^{2} \cdot 5} + 4 \sqrt{12}$

Pull terms out from under the radical.

$- (| 2 | \sqrt{5}) + 4 \sqrt{12}$

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

$- (2 \sqrt{5}) + 4 \sqrt{12}$

Multiply $2$ by $- 1$ .

$- 2 \sqrt{5} + 4 \sqrt{12}$

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

Factor $4$ out of $12$ .

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

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

$- 2 \sqrt{5} + 4 \sqrt{2^{2} \cdot 3}$

Pull terms out from under the radical.

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

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

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

Multiply $2$ by $4$ .

$- 2 \sqrt{5} + 8 \sqrt{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$- 2 \sqrt{5} + 8 \sqrt{3}$

Decimal Form:

$9.38427050 \dots$

Simplify the Radical Expression - square root of 2( square root of 10-4 square root of 6)

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