Simplify 2 square root of 30*( square root of 5+ square root of 6+ square root of 10+ square root of 15)

$2 \sqrt{30} \cdot (\sqrt{5} + \sqrt{6} + \sqrt{10} + \sqrt{15})$

Step 1

Apply the distributive property.

$2 \sqrt{30} \sqrt{5} + 2 \sqrt{30} \sqrt{6} + 2 \sqrt{30} \sqrt{10} + 2 \sqrt{30} \sqrt{15}$

Step 2

Simplify.

Multiply $2 \sqrt{30} \sqrt{5}$ .

Combine using the product rule for radicals.

$2 \sqrt{5 \cdot 30} + 2 \sqrt{30} \sqrt{6} + 2 \sqrt{30} \sqrt{10} + 2 \sqrt{30} \sqrt{15}$

Multiply $5$ by $30$ .

$2 \sqrt{150} + 2 \sqrt{30} \sqrt{6} + 2 \sqrt{30} \sqrt{10} + 2 \sqrt{30} \sqrt{15}$

Multiply $2 \sqrt{30} \sqrt{6}$ .

Combine using the product rule for radicals.

$2 \sqrt{150} + 2 \sqrt{6 \cdot 30} + 2 \sqrt{30} \sqrt{10} + 2 \sqrt{30} \sqrt{15}$

Multiply $6$ by $30$ .

$2 \sqrt{150} + 2 \sqrt{180} + 2 \sqrt{30} \sqrt{10} + 2 \sqrt{30} \sqrt{15}$

Multiply $2 \sqrt{30} \sqrt{10}$ .

Combine using the product rule for radicals.

$2 \sqrt{150} + 2 \sqrt{180} + 2 \sqrt{10 \cdot 30} + 2 \sqrt{30} \sqrt{15}$

Multiply $10$ by $30$ .

$2 \sqrt{150} + 2 \sqrt{180} + 2 \sqrt{300} + 2 \sqrt{30} \sqrt{15}$

Multiply $2 \sqrt{30} \sqrt{15}$ .

Combine using the product rule for radicals.

$2 \sqrt{150} + 2 \sqrt{180} + 2 \sqrt{300} + 2 \sqrt{15 \cdot 30}$

Multiply $15$ by $30$ .

$2 \sqrt{150} + 2 \sqrt{180} + 2 \sqrt{300} + 2 \sqrt{450}$

Step 3

Simplify each term.

Rewrite $150$ as $5^{2} \cdot 6$ .

Factor $25$ out of $150$ .

$2 \sqrt{25 (6)} + 2 \sqrt{180} + 2 \sqrt{300} + 2 \sqrt{450}$

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

$2 \sqrt{5^{2} \cdot 6} + 2 \sqrt{180} + 2 \sqrt{300} + 2 \sqrt{450}$

Pull terms out from under the radical.

$2 (5 \sqrt{6}) + 2 \sqrt{180} + 2 \sqrt{300} + 2 \sqrt{450}$

Multiply $5$ by $2$ .

$10 \sqrt{6} + 2 \sqrt{180} + 2 \sqrt{300} + 2 \sqrt{450}$

Rewrite $180$ as $6^{2} \cdot 5$ .

Factor $36$ out of $180$ .

$10 \sqrt{6} + 2 \sqrt{36 (5)} + 2 \sqrt{300} + 2 \sqrt{450}$

Rewrite $36$ as $6^{2}$ .

$10 \sqrt{6} + 2 \sqrt{6^{2} \cdot 5} + 2 \sqrt{300} + 2 \sqrt{450}$

Pull terms out from under the radical.

$10 \sqrt{6} + 2 (6 \sqrt{5}) + 2 \sqrt{300} + 2 \sqrt{450}$

Multiply $6$ by $2$ .

$10 \sqrt{6} + 12 \sqrt{5} + 2 \sqrt{300} + 2 \sqrt{450}$

Rewrite $300$ as $10^{2} \cdot 3$ .

Factor $100$ out of $300$ .

$10 \sqrt{6} + 12 \sqrt{5} + 2 \sqrt{100 (3)} + 2 \sqrt{450}$

Rewrite $100$ as $10^{2}$ .

$10 \sqrt{6} + 12 \sqrt{5} + 2 \sqrt{10^{2} \cdot 3} + 2 \sqrt{450}$

Pull terms out from under the radical.

$10 \sqrt{6} + 12 \sqrt{5} + 2 (10 \sqrt{3}) + 2 \sqrt{450}$

Multiply $10$ by $2$ .

$10 \sqrt{6} + 12 \sqrt{5} + 20 \sqrt{3} + 2 \sqrt{450}$

Rewrite $450$ as $15^{2} \cdot 2$ .

Factor $225$ out of $450$ .

$10 \sqrt{6} + 12 \sqrt{5} + 20 \sqrt{3} + 2 \sqrt{225 (2)}$

Rewrite $225$ as $15^{2}$ .

$10 \sqrt{6} + 12 \sqrt{5} + 20 \sqrt{3} + 2 \sqrt{15^{2} \cdot 2}$

Pull terms out from under the radical.

$10 \sqrt{6} + 12 \sqrt{5} + 20 \sqrt{3} + 2 (15 \sqrt{2})$

Multiply $15$ by $2$ .

$10 \sqrt{6} + 12 \sqrt{5} + 20 \sqrt{3} + 30 \sqrt{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$10 \sqrt{6} + 12 \sqrt{5} + 20 \sqrt{3} + 30 \sqrt{2}$

Decimal Form:

$128.39513618 \dots$

Simplify 2 square root of 30*( square root of 5+ square root of 6+ square root of 10+ square root of 15)

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