Solve frac{sqrt{6}-3sqrt{2}}{sqrt{2}}

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

Evaluate

$$\sqrt{3}-3\approx -1.267949192$$

Solution Steps

Rationalize the denominator of \frac{\sqrt{6}-3\sqrt{2}}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

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

The square of \sqrt{2} is 2.

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

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

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

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}.

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

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

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

The square of \sqrt{2} is 2.

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

Multiply -3 and 2 to get -6.

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

Divide each term of 2\sqrt{3}-6 by 2 to get \sqrt{3}-3.

$$\sqrt{3}-3$$