Solve 2sqrt{8}+sqrt{12}-sqrt{50}-sqrt{27/4}

$$2 \sqrt{ 8 } + \sqrt{ 12 } - \sqrt{ 50 } - \sqrt{ \frac{ 27 }{ 4 } }$$

Evaluate

$$\frac{\sqrt{3}}{2}-\sqrt{2}\approx -0.548188159$$

Short Solution Steps

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

$$2\times 2\sqrt{2}+\sqrt{12}-\sqrt{50}-\sqrt{\frac{27}{4}}$$

Multiply 2 and 2 to get 4.

$$4\sqrt{2}+\sqrt{12}-\sqrt{50}-\sqrt{\frac{27}{4}}$$

Factor 12=2^{2}\times 3. Rewrite the square root of the product \sqrt{2^{2}\times 3} as the product of square roots \sqrt{2^{2}}\sqrt{3}. Take the square root of 2^{2}.

$$4\sqrt{2}+2\sqrt{3}-\sqrt{50}-\sqrt{\frac{27}{4}}$$

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}.

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

Combine 4\sqrt{2} and -5\sqrt{2} to get -\sqrt{2}.

$$-\sqrt{2}+2\sqrt{3}-\sqrt{\frac{27}{4}}$$

Rewrite the square root of the division \sqrt{\frac{27}{4}} as the division of square roots \frac{\sqrt{27}}{\sqrt{4}}.

$$-\sqrt{2}+2\sqrt{3}-\frac{\sqrt{27}}{\sqrt{4}}$$

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}.

$$-\sqrt{2}+2\sqrt{3}-\frac{3\sqrt{3}}{\sqrt{4}}$$

Calculate the square root of 4 and get 2.

$$-\sqrt{2}+2\sqrt{3}-\frac{3\sqrt{3}}{2}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -\sqrt{2}+2\sqrt{3} times \frac{2}{2}.

$$\frac{2\left(-\sqrt{2}+2\sqrt{3}\right)}{2}-\frac{3\sqrt{3}}{2}$$

Since \frac{2\left(-\sqrt{2}+2\sqrt{3}\right)}{2} and \frac{3\sqrt{3}}{2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2\left(-\sqrt{2}+2\sqrt{3}\right)-3\sqrt{3}}{2}$$

Do the multiplications in 2\left(-\sqrt{2}+2\sqrt{3}\right)-3\sqrt{3}.

$$\frac{-2\sqrt{2}+4\sqrt{3}-3\sqrt{3}}{2}$$

Do the calculations in -2\sqrt{2}+4\sqrt{3}-3\sqrt{3}.

$$\frac{-2\sqrt{2}+\sqrt{3}}{2}$$