Solve 2sqrt{18}+sqrt{108}-sqrt{300}-sqrt{50}+sqrt{75}

$$2 \sqrt { 18 } + \sqrt { 108 } - \sqrt { 300 } - \sqrt { 50 } + \sqrt { 75 }$$

Evaluate

$$\sqrt{2}+\sqrt{3}\approx 3.14626437$$

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}.

$$2\times 3\sqrt{2}+\sqrt{108}-\sqrt{300}-\sqrt{50}+\sqrt{75}$$

Multiply 2 and 3 to get 6.

$$6\sqrt{2}+\sqrt{108}-\sqrt{300}-\sqrt{50}+\sqrt{75}$$

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}.

$$6\sqrt{2}+6\sqrt{3}-\sqrt{300}-\sqrt{50}+\sqrt{75}$$

Factor 300=10^{2}\times 3. Rewrite the square root of the product \sqrt{10^{2}\times 3} as the product of square roots \sqrt{10^{2}}\sqrt{3}. Take the square root of 10^{2}.

$$6\sqrt{2}+6\sqrt{3}-10\sqrt{3}-\sqrt{50}+\sqrt{75}$$

Combine 6\sqrt{3} and -10\sqrt{3} to get -4\sqrt{3}.

$$6\sqrt{2}-4\sqrt{3}-\sqrt{50}+\sqrt{75}$$

Factor 50=5^{2}\times 2. Rewrite the square root of the product \sqrt{5^{2}\times 2} as the product of square roots \sqrt{5^{2}}\sqrt{2}. Take the square root of 5^{2}.

$$6\sqrt{2}-4\sqrt{3}-5\sqrt{2}+\sqrt{75}$$

Combine 6\sqrt{2} and -5\sqrt{2} to get \sqrt{2}.

$$\sqrt{2}-4\sqrt{3}+\sqrt{75}$$

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}.

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

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

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