Solve sqrt{18}-(sqrt{80}+sqrt{32}-sqrt{20})

$$\sqrt { 18 } - ( \sqrt { 80 } + \sqrt { 32 } - \sqrt { 20 } )$$

Evaluate

$$-\sqrt{2}-2\sqrt{5}\approx -5.886349517$$

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}.

$$3\sqrt{2}-\left(\sqrt{80}+\sqrt{32}-\sqrt{20}\right)$$

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}.

$$3\sqrt{2}-\left(4\sqrt{5}+\sqrt{32}-\sqrt{20}\right)$$

Factor 32=4^{2}\times 2. Rewrite the square root of the product \sqrt{4^{2}\times 2} as the product of square roots \sqrt{4^{2}}\sqrt{2}. Take the square root of 4^{2}.

$$3\sqrt{2}-\left(4\sqrt{5}+4\sqrt{2}-\sqrt{20}\right)$$

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}.

$$3\sqrt{2}-\left(4\sqrt{5}+4\sqrt{2}-2\sqrt{5}\right)$$

Combine 4\sqrt{5} and -2\sqrt{5} to get 2\sqrt{5}.

$$3\sqrt{2}-\left(2\sqrt{5}+4\sqrt{2}\right)$$

To find the opposite of 2\sqrt{5}+4\sqrt{2}, find the opposite of each term.

$$3\sqrt{2}-2\sqrt{5}-4\sqrt{2}$$

Combine 3\sqrt{2} and -4\sqrt{2} to get -\sqrt{2}.

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