Evaluate -2 square root of 15(-3 square root of 3+3 square root of 5)

$- 2 \sqrt{15} (- 3 \sqrt{3} + 3 \sqrt{5})$

Step 1

Apply the distributive property.

$- 2 \sqrt{15} (- 3 \sqrt{3}) - 2 \sqrt{15} (3 \sqrt{5})$

Step 2

Multiply $- 2 \sqrt{15} (- 3 \sqrt{3})$ .

Multiply $- 3$ by $- 2$ .

$6 \sqrt{15} \sqrt{3} - 2 \sqrt{15} (3 \sqrt{5})$

Combine using the product rule for radicals.

$6 \sqrt{3 \cdot 15} - 2 \sqrt{15} (3 \sqrt{5})$

Multiply $3$ by $15$ .

$6 \sqrt{45} - 2 \sqrt{15} (3 \sqrt{5})$

$6 \sqrt{45} - 2 \sqrt{15} (3 \sqrt{5})$

Step 3

Multiply $- 2 \sqrt{15} (3 \sqrt{5})$ .

Multiply $3$ by $- 2$ .

$6 \sqrt{45} - 6 \sqrt{15} \sqrt{5}$

Combine using the product rule for radicals.

$6 \sqrt{45} - 6 \sqrt{5 \cdot 15}$

Multiply $5$ by $15$ .

$6 \sqrt{45} - 6 \sqrt{75}$

$6 \sqrt{45} - 6 \sqrt{75}$

Step 4

Simplify each term.

Rewrite $45$ as $3^{2} \cdot 5$ .

Factor $9$ out of $45$ .

$6 \sqrt{9 (5)} - 6 \sqrt{75}$

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

$6 \sqrt{3^{2} \cdot 5} - 6 \sqrt{75}$

$6 \sqrt{3^{2} \cdot 5} - 6 \sqrt{75}$

Pull terms out from under the radical.

$6 (3 \sqrt{5}) - 6 \sqrt{75}$

Multiply $3$ by $6$ .

$18 \sqrt{5} - 6 \sqrt{75}$

Rewrite $75$ as $5^{2} \cdot 3$ .

Factor $25$ out of $75$ .

$18 \sqrt{5} - 6 \sqrt{25 (3)}$

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

$18 \sqrt{5} - 6 \sqrt{5^{2} \cdot 3}$

$18 \sqrt{5} - 6 \sqrt{5^{2} \cdot 3}$

Pull terms out from under the radical.

$18 \sqrt{5} - 6 (5 \sqrt{3})$

Multiply $5$ by $- 6$ .

$18 \sqrt{5} - 30 \sqrt{3}$

$18 \sqrt{5} - 30 \sqrt{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$18 \sqrt{5} - 30 \sqrt{3}$

Decimal Form:

$- 11.71230063 \dots$

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