Solve 5-sqrt{27}+1+sqrt{75}+4-sqrt{108}

$$5- \sqrt{ 27 } +1+ \sqrt{ 75 } +4- \sqrt{ 108 }$$

Evaluate

$$10-4\sqrt{3}\approx 3.07179677$$

Solution Steps

Factor 27=3^{2}\times 3. Rewrite the square root of the product \sqrt{3^{2}\times 3} as the product of square roots \sqrt{3^{2}}\sqrt{3}. Take the square root of 3^{2}.

$$5-3\sqrt{3}+1+\sqrt{75}+4-\sqrt{108}$$

Add 5 and 1 to get 6.

$$6-3\sqrt{3}+\sqrt{75}+4-\sqrt{108}$$

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}.

$$6-3\sqrt{3}+5\sqrt{3}+4-\sqrt{108}$$

Combine -3\sqrt{3} and 5\sqrt{3} to get 2\sqrt{3}.

$$6+2\sqrt{3}+4-\sqrt{108}$$

Add 6 and 4 to get 10.

$$10+2\sqrt{3}-\sqrt{108}$$

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}.

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

Combine 2\sqrt{3} and -6\sqrt{3} to get -4\sqrt{3}.

$$10-4\sqrt{3}$$