Solve sqrt{2}-sqrt{20}+sqrt{5}+sqrt{80}-sqrt{125}

$$\sqrt{ 2 } - \sqrt{ 20 } + \sqrt{ 5 } + \sqrt{ 80 } - \sqrt{ 125 }$$

Evaluate

$$\sqrt{2}-2\sqrt{5}\approx -3.057922393$$

Solution Steps

Factor 20=2^{2}\times 5. Rewrite the square root of the product \sqrt{2^{2}\times 5} as the product of square roots \sqrt{2^{2}}\sqrt{5}. Take the square root of 2^{2}.

$$\sqrt{2}-2\sqrt{5}+\sqrt{5}+\sqrt{80}-\sqrt{125}$$

Combine -2\sqrt{5} and \sqrt{5} to get -\sqrt{5}.

$$\sqrt{2}-\sqrt{5}+\sqrt{80}-\sqrt{125}$$

Factor 80=4^{2}\times 5. Rewrite the square root of the product \sqrt{4^{2}\times 5} as the product of square roots \sqrt{4^{2}}\sqrt{5}. Take the square root of 4^{2}.

$$\sqrt{2}-\sqrt{5}+4\sqrt{5}-\sqrt{125}$$

Combine -\sqrt{5} and 4\sqrt{5} to get 3\sqrt{5}.

$$\sqrt{2}+3\sqrt{5}-\sqrt{125}$$

Factor 125=5^{2}\times 5. Rewrite the square root of the product \sqrt{5^{2}\times 5} as the product of square roots \sqrt{5^{2}}\sqrt{5}. Take the square root of 5^{2}.

$$\sqrt{2}+3\sqrt{5}-5\sqrt{5}$$

Combine 3\sqrt{5} and -5\sqrt{5} to get -2\sqrt{5}.

$$\sqrt{2}-2\sqrt{5}$$