Solve {l}{(1/3sqrt{27}-sqrt{24}}{-3sqrt{2/3})*sqrt{12}}

$$\left. \begin{array} { l } { ( \frac { 1 } { 3 } \sqrt { 27 } - \sqrt { 24 } } \\ { - 3 \sqrt { \frac { 2 } { 3 } } ) \times \sqrt { 12 } } \end{array} \right.$$

Evaluate

$$6-18\sqrt{2}\approx -19.455844123$$

Solution Steps

Factor 27=3^{2}\times 3. Rewrite the square root of the product \sqrt{3^{2}\times 3} as the product of square roots \sqrt{3^{2}}\sqrt{3}. Take the square root of 3^{2}.

$$\left(\frac{1}{3}\times 3\sqrt{3}-\sqrt{24}-3\sqrt{\frac{2}{3}}\right)\sqrt{12}$$

Cancel out 3 and 3.

$$\left(\sqrt{3}-\sqrt{24}-3\sqrt{\frac{2}{3}}\right)\sqrt{12}$$

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}.

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

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

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

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

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

The square of \sqrt{3} is 3.

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

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

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

Cancel out 3 and 3.

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

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

$$\left(\sqrt{3}-3\sqrt{6}\right)\sqrt{12}$$

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}.

$$\left(\sqrt{3}-3\sqrt{6}\right)\times 2\sqrt{3}$$

Use the distributive property to multiply \sqrt{3}-3\sqrt{6} by 2.

$$\left(2\sqrt{3}-6\sqrt{6}\right)\sqrt{3}$$

Use the distributive property to multiply 2\sqrt{3}-6\sqrt{6} by \sqrt{3}.

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

The square of \sqrt{3} is 3.

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

Multiply 2 and 3 to get 6.

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

Factor 6=3\times 2. Rewrite the square root of the product \sqrt{3\times 2} as the product of square roots \sqrt{3}\sqrt{2}.

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

Multiply \sqrt{3} and \sqrt{3} to get 3.

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

Multiply -6 and 3 to get -18.

$$6-18\sqrt{2}$$

Factor

$$6 {(1 - 3 \sqrt{2})} = -19.455844123$$