Simplify square root of 0.32*(2 square root of 800-3 square root of 1800)

$\sqrt{0.32} \cdot (2 \sqrt{800} - 3 \sqrt{1800})$

Step 1

Simplify each term.

Rewrite $800$ as $20^{2} \cdot 2$ .

Factor $400$ out of $800$ .

$\sqrt{0.32} \cdot (2 \sqrt{400 (2)} - 3 \sqrt{1800})$

Rewrite $400$ as $20^{2}$ .

$\sqrt{0.32} \cdot (2 \sqrt{20^{2} \cdot 2} - 3 \sqrt{1800})$

Pull terms out from under the radical.

$\sqrt{0.32} \cdot (2 (20 \sqrt{2}) - 3 \sqrt{1800})$

Multiply $20$ by $2$ .

$\sqrt{0.32} \cdot (40 \sqrt{2} - 3 \sqrt{1800})$

Rewrite $1800$ as $30^{2} \cdot 2$ .

Factor $900$ out of $1800$ .

$\sqrt{0.32} \cdot (40 \sqrt{2} - 3 \sqrt{900 (2)})$

Rewrite $900$ as $30^{2}$ .

$\sqrt{0.32} \cdot (40 \sqrt{2} - 3 \sqrt{30^{2} \cdot 2})$

Pull terms out from under the radical.

$\sqrt{0.32} \cdot (40 \sqrt{2} - 3 (30 \sqrt{2}))$

Multiply $30$ by $- 3$ .

$\sqrt{0.32} \cdot (40 \sqrt{2} - 90 \sqrt{2})$

Step 2

Subtract $90 \sqrt{2}$ from $40 \sqrt{2}$ .

$\sqrt{0.32} \cdot (- 50 \sqrt{2})$

Step 3

Multiply $\sqrt{0.32} (- 50 \sqrt{2})$ .

Combine using the product rule for radicals.

$- 50 \sqrt{0.32 \cdot 2}$

Multiply $0.32$ by $2$ .

$- 50 \sqrt{0.64}$

Step 4

Rewrite $0.64$ as ${0.8}^{2}$ .

$- 50 \sqrt{{0.8}^{2}}$

Step 5

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

$- 50 \cdot 0.8$

Step 6

Multiply $- 50$ by $0.8$ .

$- 40$