Solve sqrt{1000}+0,5sqrt{24}-sqrt{150}-sqrt{54}-3sqrt{40}+7sqrt{6}

$$\sqrt { 1000 } + 0,5 \sqrt { 24 } - \sqrt { 150 } - \sqrt { 54 } - 3 \sqrt { 40 } + 7 \sqrt { 6 }$$

Evaluate

$$4\sqrt{10}\approx 12.649110641$$

Solution Steps

Factor 1000=10^{2}\times 10. Rewrite the square root of the product \sqrt{10^{2}\times 10} as the product of square roots \sqrt{10^{2}}\sqrt{10}. Take the square root of 10^{2}.

$$10\sqrt{10}+0,5\sqrt{24}-\sqrt{150}-\sqrt{54}-3\sqrt{40}+7\sqrt{6}$$

Factor 24=2^{2}\times 6. Rewrite the square root of the product \sqrt{2^{2}\times 6} as the product of square roots \sqrt{2^{2}}\sqrt{6}. Take the square root of 2^{2}.

$$10\sqrt{10}+0,5\times 2\sqrt{6}-\sqrt{150}-\sqrt{54}-3\sqrt{40}+7\sqrt{6}$$

Multiply 0,5 and 2 to get 1.

$$10\sqrt{10}+\sqrt{6}-\sqrt{150}-\sqrt{54}-3\sqrt{40}+7\sqrt{6}$$

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}.

$$10\sqrt{10}+\sqrt{6}-5\sqrt{6}-\sqrt{54}-3\sqrt{40}+7\sqrt{6}$$

Combine \sqrt{6} and -5\sqrt{6} to get -4\sqrt{6}.

$$10\sqrt{10}-4\sqrt{6}-\sqrt{54}-3\sqrt{40}+7\sqrt{6}$$

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}.

$$10\sqrt{10}-4\sqrt{6}-3\sqrt{6}-3\sqrt{40}+7\sqrt{6}$$

Combine -4\sqrt{6} and -3\sqrt{6} to get -7\sqrt{6}.

$$10\sqrt{10}-7\sqrt{6}-3\sqrt{40}+7\sqrt{6}$$

Factor 40=2^{2}\times 10. Rewrite the square root of the product \sqrt{2^{2}\times 10} as the product of square roots \sqrt{2^{2}}\sqrt{10}. Take the square root of 2^{2}.

$$10\sqrt{10}-7\sqrt{6}-3\times 2\sqrt{10}+7\sqrt{6}$$

Multiply -3 and 2 to get -6.

$$10\sqrt{10}-7\sqrt{6}-6\sqrt{10}+7\sqrt{6}$$

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

$$4\sqrt{10}-7\sqrt{6}+7\sqrt{6}$$

Combine -7\sqrt{6} and 7\sqrt{6} to get 0.

$$4\sqrt{10}$$