Solve frac{5+sqrt{2}}{sqrt{8}}

$$\frac { 5 + \sqrt { 2 } } { \sqrt { 8 } }$$

Evaluate

$$\frac{5\sqrt{2}}{4}+\frac{1}{2}\approx 2.267766953$$

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}.

$$\frac{5+\sqrt{2}}{2\sqrt{2}}$$

Rationalize the denominator of \frac{5+\sqrt{2}}{2\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{\left(5+\sqrt{2}\right)\sqrt{2}}{2\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\left(5+\sqrt{2}\right)\sqrt{2}}{2\times 2}$$

Multiply 2 and 2 to get 4.

$$\frac{\left(5+\sqrt{2}\right)\sqrt{2}}{4}$$

Use the distributive property to multiply 5+\sqrt{2} by \sqrt{2}.

$$\frac{5\sqrt{2}+\left(\sqrt{2}\right)^{2}}{4}$$

The square of \sqrt{2} is 2.

$$\frac{5\sqrt{2}+2}{4}$$