Solve sqrt{12}-sqrt{1/3}-sqrt{11/3}

$$\sqrt { 12 } - \sqrt { \frac { 1 } { 3 } } - \sqrt { 1 \frac { 1 } { 3 } }$$

Evaluate

$$\sqrt{3}\approx 1.732050808$$

Short Solution Steps

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}.

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

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

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

Calculate the square root of 1 and get 1.

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

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

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

The square of \sqrt{3} is 3.

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

Multiply 1 and 3 to get 3.

$$2\sqrt{3}-\frac{\sqrt{3}}{3}-\sqrt{\frac{3+1}{3}}$$

Add 3 and 1 to get 4.

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

Rewrite the square root of the division \sqrt{\frac{4}{3}} as the division of square roots \frac{\sqrt{4}}{\sqrt{3}}.

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

Calculate the square root of 4 and get 2.

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

Rationalize the denominator of \frac{2}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

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

The square of \sqrt{3} is 3.

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

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

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

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

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

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

$$\frac{6\sqrt{3}-\sqrt{3}}{3}-\frac{2\sqrt{3}}{3}$$

Do the calculations in 6\sqrt{3}-\sqrt{3}.

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

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

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

Do the calculations in 5\sqrt{3}-2\sqrt{3}.

$$\frac{3\sqrt{3}}{3}$$

Cancel out 3 and 3.

$$\sqrt{3}$$