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

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

Evaluate

$$6-7\sqrt{3}\approx -6.124355653$$

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}.

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

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

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

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

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

The square of \sqrt{2} is 2.

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

Multiply 3 and 2 to get 6.

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

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}.

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

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

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

Multiply -3 and 2 to get -6.

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

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

$$6-6\sqrt{3}-3\times \frac{\sqrt{1}}{\sqrt{3}}$$

Calculate the square root of 1 and get 1.

$$6-6\sqrt{3}-3\times \frac{1}{\sqrt{3}}$$

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

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

The square of \sqrt{3} is 3.

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

Cancel out 3 and 3.

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

Combine -6\sqrt{3} and -\sqrt{3} to get -7\sqrt{3}.

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