Solve sqrt{2/3}-sqrt{216}+24sqrt{1/6}

$$\sqrt{ \frac{ 2 }{ 3 } } - \sqrt{ 216 } +24 \sqrt{ \frac{ 1 }{ 6 } }$$

Evaluate

$$-\frac{5\sqrt{6}}{3}\approx -4.082482905$$

Solution Steps

Rewrite the square root of the division \sqrt{\frac{2}{3}} as the division of square roots \frac{\sqrt{2}}{\sqrt{3}}.

$$\frac{\sqrt{2}}{\sqrt{3}}-\sqrt{216}+24\sqrt{\frac{1}{6}}$$

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

$$\frac{\sqrt{2}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}-\sqrt{216}+24\sqrt{\frac{1}{6}}$$

The square of \sqrt{3} is 3.

$$\frac{\sqrt{2}\sqrt{3}}{3}-\sqrt{216}+24\sqrt{\frac{1}{6}}$$

To multiply \sqrt{2} and \sqrt{3}, multiply the numbers under the square root.

$$\frac{\sqrt{6}}{3}-\sqrt{216}+24\sqrt{\frac{1}{6}}$$

Factor 216=6^{2}\times 6. Rewrite the square root of the product \sqrt{6^{2}\times 6} as the product of square roots \sqrt{6^{2}}\sqrt{6}. Take the square root of 6^{2}.

$$\frac{\sqrt{6}}{3}-6\sqrt{6}+24\sqrt{\frac{1}{6}}$$

Combine \frac{\sqrt{6}}{3} and -6\sqrt{6} to get -\frac{17}{3}\sqrt{6}.

$$-\frac{17}{3}\sqrt{6}+24\sqrt{\frac{1}{6}}$$

Rewrite the square root of the division \sqrt{\frac{1}{6}} as the division of square roots \frac{\sqrt{1}}{\sqrt{6}}.

$$-\frac{17}{3}\sqrt{6}+24\times \frac{\sqrt{1}}{\sqrt{6}}$$

Calculate the square root of 1 and get 1.

$$-\frac{17}{3}\sqrt{6}+24\times \frac{1}{\sqrt{6}}$$

Rationalize the denominator of \frac{1}{\sqrt{6}} by multiplying numerator and denominator by \sqrt{6}.

$$-\frac{17}{3}\sqrt{6}+24\times \frac{\sqrt{6}}{\left(\sqrt{6}\right)^{2}}$$

The square of \sqrt{6} is 6.

$$-\frac{17}{3}\sqrt{6}+24\times \frac{\sqrt{6}}{6}$$

Cancel out 6, the greatest common factor in 24 and 6.

$$-\frac{17}{3}\sqrt{6}+4\sqrt{6}$$

Combine -\frac{17}{3}\sqrt{6} and 4\sqrt{6} to get -\frac{5}{3}\sqrt{6}.

$$-\frac{5}{3}\sqrt{6}$$