Solve [(sqrt{3}-1)divsqrt{2}divsqrt{3}]*2

$$[ ( \sqrt { 3 } - 1 ) \div \sqrt { 2 } \div \sqrt { 3 } ] \times 2$$

Evaluate

$$-\frac{\sqrt{6}}{3}+\sqrt{2}\approx 0.597716981$$

Solution Steps

Express \frac{\frac{\sqrt{3}-1}{\sqrt{2}}}{\sqrt{3}} as a single fraction.

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

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

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

Rationalize the denominator of \frac{\sqrt{3}-1}{\sqrt{6}} by multiplying numerator and denominator by \sqrt{6}.

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

The square of \sqrt{6} is 6.

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

Cancel out 6, the greatest common factor in 2 and 6.

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

Use the distributive property to multiply \sqrt{3}-1 by \sqrt{6}.

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

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}.

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

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

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