Solve (sqrt{2}-sqrt{6})*sqrt{18}-3sqrt{1/3}quadtext{(2)}sqrt{18}*sqrt{2/3}-(1-sqrt{1})

$$( \sqrt { 2 } - \sqrt { 6 } ) \times \sqrt { 18 } - 3 \sqrt { \frac { 1 } { 3 } } \quad \text { (2) } \sqrt { 18 } \times \sqrt { \frac { 2 } { 3 } } - ( 1 - \sqrt { 1 } )$$

Evaluate

$$-6\sqrt{3}-6\approx -16.392304845$$

Solution Steps

Factor 18=3^{2}\times 2. Rewrite the square root of the product \sqrt{3^{2}\times 2} as the product of square roots \sqrt{3^{2}}\sqrt{2}. Take the square root of 3^{2}.

$$\left(\sqrt{2}-\sqrt{6}\right)\times 3\sqrt{2}-3\sqrt{\frac{1}{3}}\times 2\sqrt{18}\sqrt{\frac{2}{3}}-\left(1-\sqrt{1}\right)$$

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

$$\left(3\sqrt{2}-3\sqrt{6}\right)\sqrt{2}-3\sqrt{\frac{1}{3}}\times 2\sqrt{18}\sqrt{\frac{2}{3}}-\left(1-\sqrt{1}\right)$$

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

$$3\left(\sqrt{2}\right)^{2}-3\sqrt{6}\sqrt{2}-3\sqrt{\frac{1}{3}}\times 2\sqrt{18}\sqrt{\frac{2}{3}}-\left(1-\sqrt{1}\right)$$

The square of \sqrt{2} is 2.

$$3\times 2-3\sqrt{6}\sqrt{2}-3\sqrt{\frac{1}{3}}\times 2\sqrt{18}\sqrt{\frac{2}{3}}-\left(1-\sqrt{1}\right)$$

Multiply 3 and 2 to get 6.

$$6-3\sqrt{6}\sqrt{2}-3\sqrt{\frac{1}{3}}\times 2\sqrt{18}\sqrt{\frac{2}{3}}-\left(1-\sqrt{1}\right)$$

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

$$6-3\sqrt{2}\sqrt{3}\sqrt{2}-3\sqrt{\frac{1}{3}}\times 2\sqrt{18}\sqrt{\frac{2}{3}}-\left(1-\sqrt{1}\right)$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$6-3\times 2\sqrt{3}-3\sqrt{\frac{1}{3}}\times 2\sqrt{18}\sqrt{\frac{2}{3}}-\left(1-\sqrt{1}\right)$$

Multiply -3 and 2 to get -6.

$$6-6\sqrt{3}-3\sqrt{\frac{1}{3}}\times 2\sqrt{18}\sqrt{\frac{2}{3}}-\left(1-\sqrt{1}\right)$$

To multiply \sqrt{\frac{1}{3}} and \sqrt{\frac{2}{3}}, multiply the numbers under the square root.

$$6-6\sqrt{3}-3\sqrt{\frac{2}{9}}\times 2\sqrt{18}-\left(1-\sqrt{1}\right)$$

Rewrite the square root of the division \sqrt{\frac{2}{9}} as the division of square roots \frac{\sqrt{2}}{\sqrt{9}}.

$$6-6\sqrt{3}-3\times \frac{\sqrt{2}}{\sqrt{9}}\times 2\sqrt{18}-\left(1-\sqrt{1}\right)$$

Calculate the square root of 9 and get 3.

$$6-6\sqrt{3}-3\times \frac{\sqrt{2}}{3}\times 2\sqrt{18}-\left(1-\sqrt{1}\right)$$

Multiply 3 and 2 to get 6.

$$6-6\sqrt{3}-6\times \frac{\sqrt{2}}{3}\sqrt{18}-\left(1-\sqrt{1}\right)$$

Factor 18=3^{2}\times 2. Rewrite the square root of the product \sqrt{3^{2}\times 2} as the product of square roots \sqrt{3^{2}}\sqrt{2}. Take the square root of 3^{2}.

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

Multiply 6 and 3 to get 18.

$$6-6\sqrt{3}-18\times \frac{\sqrt{2}}{3}\sqrt{2}-\left(1-\sqrt{1}\right)$$

Cancel out 3, the greatest common factor in 18 and 3.

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

Calculate the square root of 1 and get 1.

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

Subtract 1 from 1 to get 0.

$$6-6\sqrt{3}-6\sqrt{2}\sqrt{2}-0$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$6-6\sqrt{3}-6\times 2-0$$

Multiply 6 and 2 to get 12.

$$6-6\sqrt{3}-12-0$$

Subtract 12 from 6 to get -6.

$$-6-6\sqrt{3}-0$$

Multiply -1 and 0 to get 0.

$$-6-6\sqrt{3}+0$$

Add -6 and 0 to get -6.

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