Solve sqrt{150}-2sqrt{3/2}-3sqrt{2/3}+12/sqrt{6-sqrt{24}}

$$\sqrt { 150 } - 2 \sqrt { \frac { 3 } { 2 } } - 3 \sqrt { \frac { 2 } { 3 } } + \frac { 12 } { \sqrt { 6 } - \sqrt { 24 } }$$

Evaluate

$$\sqrt{6}\approx 2.449489743$$

Solution Steps

Factor 150=5^{2}\times 6. Rewrite the square root of the product \sqrt{5^{2}\times 6} as the product of square roots \sqrt{5^{2}}\sqrt{6}. Take the square root of 5^{2}.

$$5\sqrt{6}-2\sqrt{\frac{3}{2}}-3\sqrt{\frac{2}{3}}+\frac{12}{\sqrt{6}-\sqrt{24}}$$

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

$$5\sqrt{6}-2\times \frac{\sqrt{3}}{\sqrt{2}}-3\sqrt{\frac{2}{3}}+\frac{12}{\sqrt{6}-\sqrt{24}}$$

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

$$5\sqrt{6}-2\times \frac{\sqrt{3}\sqrt{2}}{\left(\sqrt{2}\right)^{2}}-3\sqrt{\frac{2}{3}}+\frac{12}{\sqrt{6}-\sqrt{24}}$$

The square of \sqrt{2} is 2.

$$5\sqrt{6}-2\times \frac{\sqrt{3}\sqrt{2}}{2}-3\sqrt{\frac{2}{3}}+\frac{12}{\sqrt{6}-\sqrt{24}}$$

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

$$5\sqrt{6}-2\times \frac{\sqrt{6}}{2}-3\sqrt{\frac{2}{3}}+\frac{12}{\sqrt{6}-\sqrt{24}}$$

Cancel out 2 and 2.

$$5\sqrt{6}-\sqrt{6}-3\sqrt{\frac{2}{3}}+\frac{12}{\sqrt{6}-\sqrt{24}}$$

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

$$4\sqrt{6}-3\sqrt{\frac{2}{3}}+\frac{12}{\sqrt{6}-\sqrt{24}}$$

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

$$4\sqrt{6}-3\times \frac{\sqrt{2}}{\sqrt{3}}+\frac{12}{\sqrt{6}-\sqrt{24}}$$

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

$$4\sqrt{6}-3\times \frac{\sqrt{2}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}+\frac{12}{\sqrt{6}-\sqrt{24}}$$

The square of \sqrt{3} is 3.

$$4\sqrt{6}-3\times \frac{\sqrt{2}\sqrt{3}}{3}+\frac{12}{\sqrt{6}-\sqrt{24}}$$

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

$$4\sqrt{6}-3\times \frac{\sqrt{6}}{3}+\frac{12}{\sqrt{6}-\sqrt{24}}$$

Cancel out 3 and 3.

$$4\sqrt{6}-\sqrt{6}+\frac{12}{\sqrt{6}-\sqrt{24}}$$

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

$$3\sqrt{6}+\frac{12}{\sqrt{6}-\sqrt{24}}$$

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}.

$$3\sqrt{6}+\frac{12}{\sqrt{6}-2\sqrt{6}}$$

Combine \sqrt{6} and -2\sqrt{6} to get -\sqrt{6}.

$$3\sqrt{6}+\frac{12}{-\sqrt{6}}$$

Rationalize the denominator of \frac{12}{-\sqrt{6}} by multiplying numerator and denominator by \sqrt{6}.

$$3\sqrt{6}+\frac{12\sqrt{6}}{-\left(\sqrt{6}\right)^{2}}$$

The square of \sqrt{6} is 6.

$$3\sqrt{6}+\frac{12\sqrt{6}}{-6}$$

Cancel out 6 in both numerator and denominator.

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

Anything divided by -1 gives its opposite.

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

Combine 3\sqrt{6} and -2\sqrt{6} to get \sqrt{6}.

$$\sqrt{6}$$