Solve 2sqrt{5}-3sqrt{27}+2sqrt{80}-4sqrt{75}

$$2 \sqrt{ 5 } -3 \sqrt{ 27 } +2 \sqrt{ 80 } -4 \sqrt{ 75 }$$

Evaluate

$$10\sqrt{5}-29\sqrt{3}\approx -27.868793644$$

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}.

$$2\sqrt{5}-3\times 3\sqrt{3}+2\sqrt{80}-4\sqrt{75}$$

Multiply -3 and 3 to get -9.

$$2\sqrt{5}-9\sqrt{3}+2\sqrt{80}-4\sqrt{75}$$

Factor 80=4^{2}\times 5. Rewrite the square root of the product \sqrt{4^{2}\times 5} as the product of square roots \sqrt{4^{2}}\sqrt{5}. Take the square root of 4^{2}.

$$2\sqrt{5}-9\sqrt{3}+2\times 4\sqrt{5}-4\sqrt{75}$$

Multiply 2 and 4 to get 8.

$$2\sqrt{5}-9\sqrt{3}+8\sqrt{5}-4\sqrt{75}$$

Combine 2\sqrt{5} and 8\sqrt{5} to get 10\sqrt{5}.

$$10\sqrt{5}-9\sqrt{3}-4\sqrt{75}$$

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}.

$$10\sqrt{5}-9\sqrt{3}-4\times 5\sqrt{3}$$

Multiply -4 and 5 to get -20.

$$10\sqrt{5}-9\sqrt{3}-20\sqrt{3}$$

Combine -9\sqrt{3} and -20\sqrt{3} to get -29\sqrt{3}.

$$10\sqrt{5}-29\sqrt{3}$$