Solve 3sqrt{2}-sqrt{18}-sqrt{48}-sqrt{108}+sqrt{75}

$$3 \sqrt{ 2 } - \sqrt{ 18 } - \sqrt{ 48 } - \sqrt{ 108 } + \sqrt{ 75 }$$

Evaluate

$$-5\sqrt{3}\approx -8.660254038$$

Solution Steps

Factor 18=3^{2}\times 2. Rewrite the square root of the product \sqrt{3^{2}\times 2} as the product of square roots \sqrt{3^{2}}\sqrt{2}. Take the square root of 3^{2}.

$$3\sqrt{2}-3\sqrt{2}-\sqrt{48}-\sqrt{108}+\sqrt{75}$$

Combine 3\sqrt{2} and -3\sqrt{2} to get 0.

$$-\sqrt{48}-\sqrt{108}+\sqrt{75}$$

Factor 48=4^{2}\times 3. Rewrite the square root of the product \sqrt{4^{2}\times 3} as the product of square roots \sqrt{4^{2}}\sqrt{3}. Take the square root of 4^{2}.

$$-4\sqrt{3}-\sqrt{108}+\sqrt{75}$$

Factor 108=6^{2}\times 3. Rewrite the square root of the product \sqrt{6^{2}\times 3} as the product of square roots \sqrt{6^{2}}\sqrt{3}. Take the square root of 6^{2}.

$$-4\sqrt{3}-6\sqrt{3}+\sqrt{75}$$

Combine -4\sqrt{3} and -6\sqrt{3} to get -10\sqrt{3}.

$$-10\sqrt{3}+\sqrt{75}$$

Factor 75=5^{2}\times 3. Rewrite the square root of the product \sqrt{5^{2}\times 3} as the product of square roots \sqrt{5^{2}}\sqrt{3}. Take the square root of 5^{2}.

$$-10\sqrt{3}+5\sqrt{3}$$

Combine -10\sqrt{3} and 5\sqrt{3} to get -5\sqrt{3}.

$$-5\sqrt{3}$$