Solve frac{sqrt{3-1}+sqrt{3}+1}{2sqrt{2}}

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

Evaluate

$$\frac{\sqrt{2}}{4}+\frac{\sqrt{6}}{4}+\frac{1}{2}\approx 1.465925826$$

Solution Steps

Subtract 1 from 3 to get 2.

$$\frac{\sqrt{2}+\sqrt{3}+1}{2\sqrt{2}}$$

Rationalize the denominator of \frac{\sqrt{2}+\sqrt{3}+1}{2\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

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

The square of \sqrt{2} is 2.

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

Multiply 2 and 2 to get 4.

$$\frac{\left(\sqrt{2}+\sqrt{3}+1\right)\sqrt{2}}{4}$$

Use the distributive property to multiply \sqrt{2}+\sqrt{3}+1 by \sqrt{2}.

$$\frac{\left(\sqrt{2}\right)^{2}+\sqrt{3}\sqrt{2}+\sqrt{2}}{4}$$

The square of \sqrt{2} is 2.

$$\frac{2+\sqrt{3}\sqrt{2}+\sqrt{2}}{4}$$

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

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