Solve D*sqrt{8/125}

$$D \cdot \sqrt { \frac { 8 } { 125 } }$$

Evaluate

$$\frac{2\sqrt{10}D}{25}$$

Solution Steps

Rewrite the square root of the division \sqrt{\frac{8}{125}} as the division of square roots \frac{\sqrt{8}}{\sqrt{125}}.

$$D\times \frac{\sqrt{8}}{\sqrt{125}}$$

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}.

$$D\times \frac{2\sqrt{2}}{\sqrt{125}}$$

Factor 125=5^{2}\times 5. Rewrite the square root of the product \sqrt{5^{2}\times 5} as the product of square roots \sqrt{5^{2}}\sqrt{5}. Take the square root of 5^{2}.

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

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

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

The square of \sqrt{5} is 5.

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

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

$$D\times \frac{2\sqrt{10}}{5\times 5}$$

Multiply 5 and 5 to get 25.

$$D\times \frac{2\sqrt{10}}{25}$$

Express D\times \frac{2\sqrt{10}}{25} as a single fraction.

$$\frac{D\times 2\sqrt{10}}{25}$$

Differentiate w.r.t. D

$$\frac{2 \sqrt{10}}{25} = 0.2529822128134704$$