Solve 40/sqrt{40sqrt{200}}

$$\frac{ 40 }{ \sqrt{ 40 } \sqrt{ 200 } }$$

Evaluate

$$\frac{\sqrt{5}}{5}\approx 0.447213595$$

Solution Steps

Factor 200=40\times 5. Rewrite the square root of the product \sqrt{40\times 5} as the product of square roots \sqrt{40}\sqrt{5}.

$$\frac{40}{\sqrt{40}\sqrt{40}\sqrt{5}}$$

Multiply \sqrt{40} and \sqrt{40} to get 40.

$$\frac{40}{40\sqrt{5}}$$

Rationalize the denominator of \frac{40}{40\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{40\sqrt{5}}{40\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{40\sqrt{5}}{40\times 5}$$

Cancel out 40 in both numerator and denominator.

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