Solve (sqrt{24}-sqrt{1/3})-3*(sqrt{1/21}+sqrt{6})

$$( \sqrt { 24 } - \sqrt { \frac { 1 } { 3 } } ) - 3 \times ( \sqrt { \frac { 1 } { 21 } } + \sqrt { 6 } )$$

Evaluate

$$-\frac{\sqrt{3}}{3}-\frac{\sqrt{21}}{7}-\sqrt{6}\approx -3.681493683$$

Short Solution Steps

Factor 24=2^{2}\times 6. Rewrite the square root of the product \sqrt{2^{2}\times 6} as the product of square roots \sqrt{2^{2}}\sqrt{6}. Take the square root of 2^{2}.

$$2\sqrt{6}-\sqrt{\frac{1}{3}}-3\left(\sqrt{\frac{1}{21}}+\sqrt{6}\right)$$

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

$$2\sqrt{6}-\frac{\sqrt{1}}{\sqrt{3}}-3\left(\sqrt{\frac{1}{21}}+\sqrt{6}\right)$$

Calculate the square root of 1 and get 1.

$$2\sqrt{6}-\frac{1}{\sqrt{3}}-3\left(\sqrt{\frac{1}{21}}+\sqrt{6}\right)$$

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

$$2\sqrt{6}-\frac{\sqrt{3}}{\left(\sqrt{3}\right)^{2}}-3\left(\sqrt{\frac{1}{21}}+\sqrt{6}\right)$$

The square of \sqrt{3} is 3.

$$2\sqrt{6}-\frac{\sqrt{3}}{3}-3\left(\sqrt{\frac{1}{21}}+\sqrt{6}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 2\sqrt{6} times \frac{3}{3}.

$$\frac{3\times 2\sqrt{6}}{3}-\frac{\sqrt{3}}{3}-3\left(\sqrt{\frac{1}{21}}+\sqrt{6}\right)$$

Since \frac{3\times 2\sqrt{6}}{3} and \frac{\sqrt{3}}{3} have the same denominator, subtract them by subtracting their numerators.

$$\frac{3\times 2\sqrt{6}-\sqrt{3}}{3}-3\left(\sqrt{\frac{1}{21}}+\sqrt{6}\right)$$

Do the multiplications in 3\times 2\sqrt{6}-\sqrt{3}.

$$\frac{6\sqrt{6}-\sqrt{3}}{3}-3\left(\sqrt{\frac{1}{21}}+\sqrt{6}\right)$$

Rewrite the square root of the division \sqrt{\frac{1}{21}} as the division of square roots \frac{\sqrt{1}}{\sqrt{21}}.

$$\frac{6\sqrt{6}-\sqrt{3}}{3}-3\left(\frac{\sqrt{1}}{\sqrt{21}}+\sqrt{6}\right)$$

Calculate the square root of 1 and get 1.

$$\frac{6\sqrt{6}-\sqrt{3}}{3}-3\left(\frac{1}{\sqrt{21}}+\sqrt{6}\right)$$

Rationalize the denominator of \frac{1}{\sqrt{21}} by multiplying numerator and denominator by \sqrt{21}.

$$\frac{6\sqrt{6}-\sqrt{3}}{3}-3\left(\frac{\sqrt{21}}{\left(\sqrt{21}\right)^{2}}+\sqrt{6}\right)$$

The square of \sqrt{21} is 21.

$$\frac{6\sqrt{6}-\sqrt{3}}{3}-3\left(\frac{\sqrt{21}}{21}+\sqrt{6}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Multiply \sqrt{6} times \frac{21}{21}.

$$\frac{6\sqrt{6}-\sqrt{3}}{3}-3\left(\frac{\sqrt{21}}{21}+\frac{21\sqrt{6}}{21}\right)$$

Since \frac{\sqrt{21}}{21} and \frac{21\sqrt{6}}{21} have the same denominator, add them by adding their numerators.

$$\frac{6\sqrt{6}-\sqrt{3}}{3}-3\times \frac{\sqrt{21}+21\sqrt{6}}{21}$$

Cancel out 21, the greatest common factor in 3 and 21.

$$\frac{6\sqrt{6}-\sqrt{3}}{3}-\frac{\sqrt{21}+21\sqrt{6}}{7}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3 and 7 is 21. Multiply \frac{6\sqrt{6}-\sqrt{3}}{3} times \frac{7}{7}. Multiply \frac{\sqrt{21}+21\sqrt{6}}{7} times \frac{3}{3}.

$$\frac{7\left(6\sqrt{6}-\sqrt{3}\right)}{21}-\frac{3\left(\sqrt{21}+21\sqrt{6}\right)}{21}$$

Since \frac{7\left(6\sqrt{6}-\sqrt{3}\right)}{21} and \frac{3\left(\sqrt{21}+21\sqrt{6}\right)}{21} have the same denominator, subtract them by subtracting their numerators.

$$\frac{7\left(6\sqrt{6}-\sqrt{3}\right)-3\left(\sqrt{21}+21\sqrt{6}\right)}{21}$$

Do the multiplications in 7\left(6\sqrt{6}-\sqrt{3}\right)-3\left(\sqrt{21}+21\sqrt{6}\right).

$$\frac{42\sqrt{6}-7\sqrt{3}-3\sqrt{21}-63\sqrt{6}}{21}$$

Do the calculations in 42\sqrt{6}-7\sqrt{3}-3\sqrt{21}-63\sqrt{6}.

$$\frac{-21\sqrt{6}-7\sqrt{3}-3\sqrt{21}}{21}$$