Solve 2/3sqrt{1/3}+1/2sqrt{48}-sqrt{75/4}

$$\frac{ 2 }{ 3 } \sqrt{ \frac{ 1 }{ 3 } } + \frac{ 1 }{ 2 } \sqrt{ 48 } - \sqrt{ \frac{ 75 }{ 4 } }$$

Evaluate

$$-\frac{5\sqrt{3}}{18}\approx -0.481125224$$

Short Solution Steps

Rewrite the square root of the division \sqrt{\frac{1}{3}} as the division of square roots \frac{\sqrt{1}}{\sqrt{3}}.

$$\frac{2}{3}\times \frac{\sqrt{1}}{\sqrt{3}}+\frac{1}{2}\sqrt{48}-\sqrt{\frac{75}{4}}$$

Calculate the square root of 1 and get 1.

$$\frac{2}{3}\times \frac{1}{\sqrt{3}}+\frac{1}{2}\sqrt{48}-\sqrt{\frac{75}{4}}$$

Rationalize the denominator of \frac{1}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{2}{3}\times \frac{\sqrt{3}}{\left(\sqrt{3}\right)^{2}}+\frac{1}{2}\sqrt{48}-\sqrt{\frac{75}{4}}$$

The square of \sqrt{3} is 3.

$$\frac{2}{3}\times \frac{\sqrt{3}}{3}+\frac{1}{2}\sqrt{48}-\sqrt{\frac{75}{4}}$$

Multiply \frac{2}{3} times \frac{\sqrt{3}}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{2\sqrt{3}}{3\times 3}+\frac{1}{2}\sqrt{48}-\sqrt{\frac{75}{4}}$$

Factor 48=4^{2}\times 3. Rewrite the square root of the product \sqrt{4^{2}\times 3} as the product of square roots \sqrt{4^{2}}\sqrt{3}. Take the square root of 4^{2}.

$$\frac{2\sqrt{3}}{3\times 3}+\frac{1}{2}\times 4\sqrt{3}-\sqrt{\frac{75}{4}}$$

Multiply \frac{1}{2} and 4 to get \frac{4}{2}.

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

Divide 4 by 2 to get 2.

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

Rewrite the square root of the division \sqrt{\frac{75}{4}} as the division of square roots \frac{\sqrt{75}}{\sqrt{4}}.

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

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}.

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

Calculate the square root of 4 and get 2.

$$\frac{2\sqrt{3}}{3\times 3}+2\sqrt{3}-\frac{5\sqrt{3}}{2}$$

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

$$\frac{2\sqrt{3}}{3\times 3}+\frac{2\sqrt{3}\times 3\times 3}{3\times 3}-\frac{5\sqrt{3}}{2}$$

Since \frac{2\sqrt{3}}{3\times 3} and \frac{2\sqrt{3}\times 3\times 3}{3\times 3} have the same denominator, add them by adding their numerators.

$$\frac{2\sqrt{3}+2\sqrt{3}\times 3\times 3}{3\times 3}-\frac{5\sqrt{3}}{2}$$

Do the multiplications in 2\sqrt{3}+2\sqrt{3}\times 3\times 3.

$$\frac{2\sqrt{3}+18\sqrt{3}}{3\times 3}-\frac{5\sqrt{3}}{2}$$

Do the calculations in 2\sqrt{3}+18\sqrt{3}.

$$\frac{20\sqrt{3}}{3\times 3}-\frac{5\sqrt{3}}{2}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3\times 3 and 2 is 18. Multiply \frac{20\sqrt{3}}{3\times 3} times \frac{2}{2}. Multiply \frac{5\sqrt{3}}{2} times \frac{9}{9}.

$$\frac{2\times 20\sqrt{3}}{18}-\frac{9\times 5\sqrt{3}}{18}$$

Since \frac{2\times 20\sqrt{3}}{18} and \frac{9\times 5\sqrt{3}}{18} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2\times 20\sqrt{3}-9\times 5\sqrt{3}}{18}$$

Do the multiplications in 2\times 20\sqrt{3}-9\times 5\sqrt{3}.

$$\frac{40\sqrt{3}-45\sqrt{3}}{18}$$

Do the calculations in 40\sqrt{3}-45\sqrt{3}.

$$\frac{-5\sqrt{3}}{18}$$