Solve {left(3/2sqrt{5/3}right)}^+sqrt{5/4}

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

Evaluate

$$\frac{\sqrt{5}+\sqrt{15}}{2}\approx 3.054525662$$

Solution Steps

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

$$\left(\frac{3}{2}\times \frac{\sqrt{5}}{\sqrt{3}}\right)^{1}+\sqrt{\frac{5}{4}}$$

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

$$\left(\frac{3}{2}\times \frac{\sqrt{5}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}\right)^{1}+\sqrt{\frac{5}{4}}$$

The square of \sqrt{3} is 3.

$$\left(\frac{3}{2}\times \frac{\sqrt{5}\sqrt{3}}{3}\right)^{1}+\sqrt{\frac{5}{4}}$$

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

$$\left(\frac{3}{2}\times \frac{\sqrt{15}}{3}\right)^{1}+\sqrt{\frac{5}{4}}$$

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

$$\left(\frac{3\sqrt{15}}{2\times 3}\right)^{1}+\sqrt{\frac{5}{4}}$$

Cancel out 3 in both numerator and denominator.

$$\left(\frac{\sqrt{15}}{2}\right)^{1}+\sqrt{\frac{5}{4}}$$

Calculate \frac{\sqrt{15}}{2} to the power of 1 and get \frac{\sqrt{15}}{2}.

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

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, add them by adding their numerators.

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

Factor

$$\frac{\sqrt{5} + \sqrt{15}}{2} = 3.0545256618536034$$