Solve frac{sqrt{3}-sqrt{2}}{sqrt{5}}*6/sqrt{6}

$$\frac { \sqrt { 3 } - \sqrt { 2 } } { \sqrt { 5 } } \times \frac { 6 } { \sqrt { 6 } }$$

Evaluate

$$\frac{3\sqrt{10}-2\sqrt{15}}{5}\approx 0.348173258$$

Solution Steps

Rationalize the denominator of \frac{\sqrt{3}-\sqrt{2}}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{\left(\sqrt{3}-\sqrt{2}\right)\sqrt{5}}{\left(\sqrt{5}\right)^{2}}\times \frac{6}{\sqrt{6}}$$

The square of \sqrt{5} is 5.

$$\frac{\left(\sqrt{3}-\sqrt{2}\right)\sqrt{5}}{5}\times \frac{6}{\sqrt{6}}$$

Rationalize the denominator of \frac{6}{\sqrt{6}} by multiplying numerator and denominator by \sqrt{6}.

$$\frac{\left(\sqrt{3}-\sqrt{2}\right)\sqrt{5}}{5}\times \frac{6\sqrt{6}}{\left(\sqrt{6}\right)^{2}}$$

The square of \sqrt{6} is 6.

$$\frac{\left(\sqrt{3}-\sqrt{2}\right)\sqrt{5}}{5}\times \frac{6\sqrt{6}}{6}$$

Cancel out 6 and 6.

$$\frac{\left(\sqrt{3}-\sqrt{2}\right)\sqrt{5}}{5}\sqrt{6}$$

Express \frac{\left(\sqrt{3}-\sqrt{2}\right)\sqrt{5}}{5}\sqrt{6} as a single fraction.

$$\frac{\left(\sqrt{3}-\sqrt{2}\right)\sqrt{5}\sqrt{6}}{5}$$

To multiply \sqrt{5} and \sqrt{6}, multiply the numbers under the square root.

$$\frac{\left(\sqrt{3}-\sqrt{2}\right)\sqrt{30}}{5}$$

Use the distributive property to multiply \sqrt{3}-\sqrt{2} by \sqrt{30}.

$$\frac{\sqrt{3}\sqrt{30}-\sqrt{2}\sqrt{30}}{5}$$

Factor 30=3\times 10. Rewrite the square root of the product \sqrt{3\times 10} as the product of square roots \sqrt{3}\sqrt{10}.

$$\frac{\sqrt{3}\sqrt{3}\sqrt{10}-\sqrt{2}\sqrt{30}}{5}$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$\frac{3\sqrt{10}-\sqrt{2}\sqrt{30}}{5}$$

Factor 30=2\times 15. Rewrite the square root of the product \sqrt{2\times 15} as the product of square roots \sqrt{2}\sqrt{15}.

$$\frac{3\sqrt{10}-\sqrt{2}\sqrt{2}\sqrt{15}}{5}$$

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

$$\frac{3\sqrt{10}-2\sqrt{15}}{5}$$