Solve sqrt{216}-2sqrt{54}-sqrt{160}+4sqrt{360}-2sqrt{50}

$$\sqrt{ 216 } -2 \sqrt{ 54 } - \sqrt{ 160 } +4 \sqrt{ 360 } -2 \sqrt{ 50 }$$

Evaluate

$$10\sqrt{2}\left(2\sqrt{5}-1\right)\approx 49.10341758$$

Solution Steps

Factor 216=6^{2}\times 6. Rewrite the square root of the product \sqrt{6^{2}\times 6} as the product of square roots \sqrt{6^{2}}\sqrt{6}. Take the square root of 6^{2}.

$$6\sqrt{6}-2\sqrt{54}-\sqrt{160}+4\sqrt{360}-2\sqrt{50}$$

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}.

$$6\sqrt{6}-2\times 3\sqrt{6}-\sqrt{160}+4\sqrt{360}-2\sqrt{50}$$

Multiply -2 and 3 to get -6.

$$6\sqrt{6}-6\sqrt{6}-\sqrt{160}+4\sqrt{360}-2\sqrt{50}$$

Combine 6\sqrt{6} and -6\sqrt{6} to get 0.

$$-\sqrt{160}+4\sqrt{360}-2\sqrt{50}$$

Factor 160=4^{2}\times 10. Rewrite the square root of the product \sqrt{4^{2}\times 10} as the product of square roots \sqrt{4^{2}}\sqrt{10}. Take the square root of 4^{2}.

$$-4\sqrt{10}+4\sqrt{360}-2\sqrt{50}$$

Factor 360=6^{2}\times 10. Rewrite the square root of the product \sqrt{6^{2}\times 10} as the product of square roots \sqrt{6^{2}}\sqrt{10}. Take the square root of 6^{2}.

$$-4\sqrt{10}+4\times 6\sqrt{10}-2\sqrt{50}$$

Multiply 4 and 6 to get 24.

$$-4\sqrt{10}+24\sqrt{10}-2\sqrt{50}$$

Combine -4\sqrt{10} and 24\sqrt{10} to get 20\sqrt{10}.

$$20\sqrt{10}-2\sqrt{50}$$

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}.

$$20\sqrt{10}-2\times 5\sqrt{2}$$

Multiply -2 and 5 to get -10.

$$20\sqrt{10}-10\sqrt{2}$$