Solve frac{sqrt{3}-sqrt{8}}{sqrt{2}}

$$\frac { \sqrt { 3 } - \sqrt { 8 } } { \sqrt { 2 } }$$

Evaluate

$$\frac{\sqrt{6}}{2}-2\approx -0.775255129$$

Solution Steps

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

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

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

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

The square of \sqrt{2} is 2.

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

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

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

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

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

The square of \sqrt{2} is 2.

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

Multiply -2 and 2 to get -4.

$$\frac{\sqrt{6}-4}{2}$$