Solve sqrt{3/4}+sqrt{27/4}+sqrt{1/2}+sqrt{25/2}

$$\sqrt { \frac { 3 } { 4 } } + \sqrt { \frac { 27 } { 4 } } + \sqrt { \frac { 1 } { 2 } } + \sqrt { \frac { 25 } { 2 } }$$

Evaluate

$$2\sqrt{3}+3\sqrt{2}\approx 7.706742302$$

Solution Steps

Rewrite the square root of the division \sqrt{\frac{3}{4}} as the division of square roots \frac{\sqrt{3}}{\sqrt{4}}.

$$\frac{\sqrt{3}}{\sqrt{4}}+\sqrt{\frac{27}{4}}+\sqrt{\frac{1}{2}}+\sqrt{\frac{25}{2}}$$

Calculate the square root of 4 and get 2.

$$\frac{\sqrt{3}}{2}+\sqrt{\frac{27}{4}}+\sqrt{\frac{1}{2}}+\sqrt{\frac{25}{2}}$$

Rewrite the square root of the division \sqrt{\frac{27}{4}} as the division of square roots \frac{\sqrt{27}}{\sqrt{4}}.

$$\frac{\sqrt{3}}{2}+\frac{\sqrt{27}}{\sqrt{4}}+\sqrt{\frac{1}{2}}+\sqrt{\frac{25}{2}}$$

Factor 27=3^{2}\times 3. Rewrite the square root of the product \sqrt{3^{2}\times 3} as the product of square roots \sqrt{3^{2}}\sqrt{3}. Take the square root of 3^{2}.

$$\frac{\sqrt{3}}{2}+\frac{3\sqrt{3}}{\sqrt{4}}+\sqrt{\frac{1}{2}}+\sqrt{\frac{25}{2}}$$

Calculate the square root of 4 and get 2.

$$\frac{\sqrt{3}}{2}+\frac{3\sqrt{3}}{2}+\sqrt{\frac{1}{2}}+\sqrt{\frac{25}{2}}$$

Combine \frac{\sqrt{3}}{2} and \frac{3\sqrt{3}}{2} to get 2\sqrt{3}.

$$2\sqrt{3}+\sqrt{\frac{1}{2}}+\sqrt{\frac{25}{2}}$$

Rewrite the square root of the division \sqrt{\frac{1}{2}} as the division of square roots \frac{\sqrt{1}}{\sqrt{2}}.

$$2\sqrt{3}+\frac{\sqrt{1}}{\sqrt{2}}+\sqrt{\frac{25}{2}}$$

Calculate the square root of 1 and get 1.

$$2\sqrt{3}+\frac{1}{\sqrt{2}}+\sqrt{\frac{25}{2}}$$

Rationalize the denominator of \frac{1}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$2\sqrt{3}+\frac{\sqrt{2}}{\left(\sqrt{2}\right)^{2}}+\sqrt{\frac{25}{2}}$$

The square of \sqrt{2} is 2.

$$2\sqrt{3}+\frac{\sqrt{2}}{2}+\sqrt{\frac{25}{2}}$$

Rewrite the square root of the division \sqrt{\frac{25}{2}} as the division of square roots \frac{\sqrt{25}}{\sqrt{2}}.

$$2\sqrt{3}+\frac{\sqrt{2}}{2}+\frac{\sqrt{25}}{\sqrt{2}}$$

Calculate the square root of 25 and get 5.

$$2\sqrt{3}+\frac{\sqrt{2}}{2}+\frac{5}{\sqrt{2}}$$

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

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

The square of \sqrt{2} is 2.

$$2\sqrt{3}+\frac{\sqrt{2}}{2}+\frac{5\sqrt{2}}{2}$$

Combine \frac{\sqrt{2}}{2} and \frac{5\sqrt{2}}{2} to get 3\sqrt{2}.

$$2\sqrt{3}+3\sqrt{2}$$