Solve frac{3sqrt{2}+2}{sqrt{8}}+sqrt{1/2}

$$\frac { 3 \sqrt { 2 } + 2 } { \sqrt { 8 } } + \sqrt { \frac { 1 } { 2 } }$$

Evaluate

$$\sqrt{2}+\frac{3}{2}\approx 2.914213562$$

Short 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}.

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

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

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

The square of \sqrt{2} is 2.

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

Multiply 2 and 2 to get 4.

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

Rewrite the square root of the division \sqrt{\frac{1}{2}} as the division of square roots \frac{\sqrt{1}}{\sqrt{2}}.

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

Calculate the square root of 1 and get 1.

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

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

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

The square of \sqrt{2} is 2.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 4 and 2 is 4. Multiply \frac{\sqrt{2}}{2} times \frac{2}{2}.

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

Since \frac{\left(3\sqrt{2}+2\right)\sqrt{2}}{4} and \frac{2\sqrt{2}}{4} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(3\sqrt{2}+2\right)\sqrt{2}+2\sqrt{2}.

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

Do the calculations in 6+2\sqrt{2}+2\sqrt{2}.

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