Solve 7sqrt{54}-3sqrt{18}+sqrt{24}-3/5sqrt{50}-sqrt{6}

$$7 \sqrt{ 54 } -3 \sqrt{ 18 } + \sqrt{ 24 } - \frac{ 3 }{ 5 } \sqrt{ 50 } - \sqrt{ 6 }$$

Evaluate

$$22\sqrt{6}-12\sqrt{2}\approx 36.918211593$$

Solution Steps

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}.

$$7\times 3\sqrt{6}-3\sqrt{18}+\sqrt{24}-\frac{3}{5}\sqrt{50}-\sqrt{6}$$

Multiply 7 and 3 to get 21.

$$21\sqrt{6}-3\sqrt{18}+\sqrt{24}-\frac{3}{5}\sqrt{50}-\sqrt{6}$$

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}.

$$21\sqrt{6}-3\times 3\sqrt{2}+\sqrt{24}-\frac{3}{5}\sqrt{50}-\sqrt{6}$$

Multiply -3 and 3 to get -9.

$$21\sqrt{6}-9\sqrt{2}+\sqrt{24}-\frac{3}{5}\sqrt{50}-\sqrt{6}$$

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}.

$$21\sqrt{6}-9\sqrt{2}+2\sqrt{6}-\frac{3}{5}\sqrt{50}-\sqrt{6}$$

Combine 21\sqrt{6} and 2\sqrt{6} to get 23\sqrt{6}.

$$23\sqrt{6}-9\sqrt{2}-\frac{3}{5}\sqrt{50}-\sqrt{6}$$

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}.

$$23\sqrt{6}-9\sqrt{2}-\frac{3}{5}\times 5\sqrt{2}-\sqrt{6}$$

Cancel out 5 and 5.

$$23\sqrt{6}-9\sqrt{2}-3\sqrt{2}-\sqrt{6}$$

Combine -9\sqrt{2} and -3\sqrt{2} to get -12\sqrt{2}.

$$23\sqrt{6}-12\sqrt{2}-\sqrt{6}$$

Combine 23\sqrt{6} and -\sqrt{6} to get 22\sqrt{6}.

$$22\sqrt{6}-12\sqrt{2}$$

Factor

$$2 {(11 \sqrt{6} - 6 \sqrt{2})} = 36.918211593$$