Solve frac{sqrt{8}*sqrt{10}}{sqrt{20}*sqrt{2}}

$$\frac { \sqrt { 8 } \cdot \sqrt { 10 } } { \sqrt { 20 } \cdot \sqrt { 2 } }$$

Evaluate

$$\sqrt{2}\approx 1.414213562$$

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}.

$$\frac{2\sqrt{2}\sqrt{10}}{\sqrt{20}\sqrt{2}}$$

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}.

$$\frac{2\sqrt{2}\sqrt{2}\sqrt{5}}{\sqrt{20}\sqrt{2}}$$

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

$$\frac{2\times 2\sqrt{5}}{\sqrt{20}\sqrt{2}}$$

Factor 20=2\times 10. Rewrite the square root of the product \sqrt{2\times 10} as the product of square roots \sqrt{2}\sqrt{10}.

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

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

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

Cancel out 2 in both numerator and denominator.

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

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

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

The square of \sqrt{10} is 10.

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

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}.

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

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

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

Multiply 2 and 5 to get 10.

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

Cancel out 10 and 10.

$$\sqrt{2}$$