Solve sqrt{392}-sqrt{150}+sqrt{578}-sqrt{486}

$$\sqrt { 392 } - \sqrt { 150 } + \sqrt { 578 } - \sqrt { 486 }$$

Evaluate

$$31\sqrt{2}-14\sqrt{6}\approx 9.547764035$$

Solution Steps

Factor 392=14^{2}\times 2. Rewrite the square root of the product \sqrt{14^{2}\times 2} as the product of square roots \sqrt{14^{2}}\sqrt{2}. Take the square root of 14^{2}.

$$14\sqrt{2}-\sqrt{150}+\sqrt{578}-\sqrt{486}$$

Factor 150=5^{2}\times 6. Rewrite the square root of the product \sqrt{5^{2}\times 6} as the product of square roots \sqrt{5^{2}}\sqrt{6}. Take the square root of 5^{2}.

$$14\sqrt{2}-5\sqrt{6}+\sqrt{578}-\sqrt{486}$$

Factor 578=17^{2}\times 2. Rewrite the square root of the product \sqrt{17^{2}\times 2} as the product of square roots \sqrt{17^{2}}\sqrt{2}. Take the square root of 17^{2}.

$$14\sqrt{2}-5\sqrt{6}+17\sqrt{2}-\sqrt{486}$$

Combine 14\sqrt{2} and 17\sqrt{2} to get 31\sqrt{2}.

$$31\sqrt{2}-5\sqrt{6}-\sqrt{486}$$

Factor 486=9^{2}\times 6. Rewrite the square root of the product \sqrt{9^{2}\times 6} as the product of square roots \sqrt{9^{2}}\sqrt{6}. Take the square root of 9^{2}.

$$31\sqrt{2}-5\sqrt{6}-9\sqrt{6}$$

Combine -5\sqrt{6} and -9\sqrt{6} to get -14\sqrt{6}.

$$31\sqrt{2}-14\sqrt{6}$$