Solve (3/2sqrt{5/3}-sqrt{5/4}

$$( \frac{ 3 }{ 2 } \sqrt{ \frac{ 5 }{ 3 } } - \sqrt{ \frac{ 5 }{ 4 } }$$

Evaluate

$$\frac{\sqrt{15}-\sqrt{5}}{2}\approx 0.818457684$$

Short Solution Steps

Rewrite the square root of the division \sqrt{\frac{5}{3}} as the division of square roots \frac{\sqrt{5}}{\sqrt{3}}.

$$\frac{3}{2}\times \frac{\sqrt{5}}{\sqrt{3}}-\sqrt{\frac{5}{4}}$$

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

$$\frac{3}{2}\times \frac{\sqrt{5}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}-\sqrt{\frac{5}{4}}$$

The square of \sqrt{3} is 3.

$$\frac{3}{2}\times \frac{\sqrt{5}\sqrt{3}}{3}-\sqrt{\frac{5}{4}}$$

To multiply \sqrt{5} and \sqrt{3}, multiply the numbers under the square root.

$$\frac{3}{2}\times \frac{\sqrt{15}}{3}-\sqrt{\frac{5}{4}}$$

Multiply \frac{3}{2} times \frac{\sqrt{15}}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{3\sqrt{15}}{2\times 3}-\sqrt{\frac{5}{4}}$$

Cancel out 3 in both numerator and denominator.

$$\frac{\sqrt{15}}{2}-\sqrt{\frac{5}{4}}$$

Rewrite the square root of the division \sqrt{\frac{5}{4}} as the division of square roots \frac{\sqrt{5}}{\sqrt{4}}.

$$\frac{\sqrt{15}}{2}-\frac{\sqrt{5}}{\sqrt{4}}$$

Calculate the square root of 4 and get 2.

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

Since \frac{\sqrt{15}}{2} and \frac{\sqrt{5}}{2} have the same denominator, subtract them by subtracting their numerators.

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

Factor

$$\frac{\sqrt{15} - \sqrt{5}}{2} = 0.8184576843538136$$