Solve frac{sqrt{40}-sqrt{2}}{sqrt{5}}

$$\frac{ \sqrt{ 40 } - \sqrt{ 2 } }{ \sqrt{ 5 } }$$

Evaluate

$$\frac{\sqrt{10}\left(2\sqrt{5}-1\right)}{5}\approx 2.195971593$$

Solution Steps

Factor 40=2^{2}\times 10. Rewrite the square root of the product \sqrt{2^{2}\times 10} as the product of square roots \sqrt{2^{2}}\sqrt{10}. Take the square root of 2^{2}.

$$\frac{2\sqrt{10}-\sqrt{2}}{\sqrt{5}}$$

Rationalize the denominator of \frac{2\sqrt{10}-\sqrt{2}}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{\left(2\sqrt{10}-\sqrt{2}\right)\sqrt{5}}{\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{\left(2\sqrt{10}-\sqrt{2}\right)\sqrt{5}}{5}$$

Use the distributive property to multiply 2\sqrt{10}-\sqrt{2} by \sqrt{5}.

$$\frac{2\sqrt{10}\sqrt{5}-\sqrt{2}\sqrt{5}}{5}$$

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

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

Multiply \sqrt{5} and \sqrt{5} to get 5.

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

Multiply 2 and 5 to get 10.

$$\frac{10\sqrt{2}-\sqrt{2}\sqrt{5}}{5}$$

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

$$\frac{10\sqrt{2}-\sqrt{10}}{5}$$