Solve sqrt{75}-sqrt{54}+sqrt{96}-sqrt{108}

$$\sqrt{ 75 } - \sqrt{ 54 } + \sqrt{ 96 } - \sqrt{ 108 }$$

Evaluate

$$\sqrt{6}-\sqrt{3}\approx 0.717438935$$

Solution Steps

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}.

$$5\sqrt{3}-\sqrt{54}+\sqrt{96}-\sqrt{108}$$

Factor 54=3^{2}\times 6. Rewrite the square root of the product \sqrt{3^{2}\times 6} as the product of square roots \sqrt{3^{2}}\sqrt{6}. Take the square root of 3^{2}.

$$5\sqrt{3}-3\sqrt{6}+\sqrt{96}-\sqrt{108}$$

Factor 96=4^{2}\times 6. Rewrite the square root of the product \sqrt{4^{2}\times 6} as the product of square roots \sqrt{4^{2}}\sqrt{6}. Take the square root of 4^{2}.

$$5\sqrt{3}-3\sqrt{6}+4\sqrt{6}-\sqrt{108}$$

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

$$5\sqrt{3}+\sqrt{6}-\sqrt{108}$$

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}.

$$5\sqrt{3}+\sqrt{6}-6\sqrt{3}$$

Combine 5\sqrt{3} and -6\sqrt{3} to get -\sqrt{3}.

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