Solve 2/sqrt{10*sqrt{8}}

$$\frac { 2 } { \sqrt { 10 } \times \sqrt { 8 } }$$

Evaluate

$$\frac{\sqrt{5}}{10}\approx 0.223606798$$

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}.

$$\frac{2}{\sqrt{10}\times 2\sqrt{2}}$$

Factor 10=2\times 5. Rewrite the square root of the product \sqrt{2\times 5} as the product of square roots \sqrt{2}\sqrt{5}.

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

Multiply \sqrt{2} and \sqrt{2} to get 2.

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

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

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

The square of \sqrt{5} is 5.

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

Cancel out 2 in both numerator and denominator.

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

Multiply 2 and 5 to get 10.

$$\frac{\sqrt{5}}{10}$$