Solve 4sqrt{75}+6sqrt{18}-sqrt{128}-sqrt{245}-sqrt{98}-3sqrt{125}

$$4 \sqrt { 75 } + 6 \sqrt { 18 } - \sqrt { 128 } - \sqrt { 245 } - \sqrt { 98 } - 3 \sqrt { 125 }$$

Evaluate

$$3\sqrt{2}+20\sqrt{3}-22\sqrt{5}\approx -10.309838666$$

Solution Steps

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}.

$$4\times 5\sqrt{3}+6\sqrt{18}-\sqrt{128}-\sqrt{245}-\sqrt{98}-3\sqrt{125}$$

Multiply 4 and 5 to get 20.

$$20\sqrt{3}+6\sqrt{18}-\sqrt{128}-\sqrt{245}-\sqrt{98}-3\sqrt{125}$$

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}.

$$20\sqrt{3}+6\times 3\sqrt{2}-\sqrt{128}-\sqrt{245}-\sqrt{98}-3\sqrt{125}$$

Multiply 6 and 3 to get 18.

$$20\sqrt{3}+18\sqrt{2}-\sqrt{128}-\sqrt{245}-\sqrt{98}-3\sqrt{125}$$

Factor 128=8^{2}\times 2. Rewrite the square root of the product \sqrt{8^{2}\times 2} as the product of square roots \sqrt{8^{2}}\sqrt{2}. Take the square root of 8^{2}.

$$20\sqrt{3}+18\sqrt{2}-8\sqrt{2}-\sqrt{245}-\sqrt{98}-3\sqrt{125}$$

Combine 18\sqrt{2} and -8\sqrt{2} to get 10\sqrt{2}.

$$20\sqrt{3}+10\sqrt{2}-\sqrt{245}-\sqrt{98}-3\sqrt{125}$$

Factor 245=7^{2}\times 5. Rewrite the square root of the product \sqrt{7^{2}\times 5} as the product of square roots \sqrt{7^{2}}\sqrt{5}. Take the square root of 7^{2}.

$$20\sqrt{3}+10\sqrt{2}-7\sqrt{5}-\sqrt{98}-3\sqrt{125}$$

Factor 98=7^{2}\times 2. Rewrite the square root of the product \sqrt{7^{2}\times 2} as the product of square roots \sqrt{7^{2}}\sqrt{2}. Take the square root of 7^{2}.

$$20\sqrt{3}+10\sqrt{2}-7\sqrt{5}-7\sqrt{2}-3\sqrt{125}$$

Combine 10\sqrt{2} and -7\sqrt{2} to get 3\sqrt{2}.

$$20\sqrt{3}+3\sqrt{2}-7\sqrt{5}-3\sqrt{125}$$

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}.

$$20\sqrt{3}+3\sqrt{2}-7\sqrt{5}-3\times 5\sqrt{5}$$

Multiply -3 and 5 to get -15.

$$20\sqrt{3}+3\sqrt{2}-7\sqrt{5}-15\sqrt{5}$$

Combine -7\sqrt{5} and -15\sqrt{5} to get -22\sqrt{5}.

$$20\sqrt{3}+3\sqrt{2}-22\sqrt{5}$$