Solve frac{2sqrt{5}+sqrt{3}}{sqrt{8}}

$$\frac { 2 \sqrt { 5 } + \sqrt { 3 } } { \sqrt { 8 } }$$

Evaluate

$$\frac{\sqrt{6}}{4}+\frac{\sqrt{10}}{2}\approx 2.193511266$$

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}.

$$\frac{2\sqrt{5}+\sqrt{3}}{2\sqrt{2}}$$

Rationalize the denominator of \frac{2\sqrt{5}+\sqrt{3}}{2\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{\left(2\sqrt{5}+\sqrt{3}\right)\sqrt{2}}{2\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\left(2\sqrt{5}+\sqrt{3}\right)\sqrt{2}}{2\times 2}$$

Multiply 2 and 2 to get 4.

$$\frac{\left(2\sqrt{5}+\sqrt{3}\right)\sqrt{2}}{4}$$

Use the distributive property to multiply 2\sqrt{5}+\sqrt{3} by \sqrt{2}.

$$\frac{2\sqrt{5}\sqrt{2}+\sqrt{3}\sqrt{2}}{4}$$

To multiply \sqrt{5} and \sqrt{2}, multiply the numbers under the square root.

$$\frac{2\sqrt{10}+\sqrt{3}\sqrt{2}}{4}$$

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

$$\frac{2\sqrt{10}+\sqrt{6}}{4}$$