Solve sqrt{14}+sqrt{2/7}+sqrt{7/2}

$$\sqrt{ 14 } + \sqrt{ \frac{ 2 }{ 7 } } + \sqrt{ \frac{ 7 }{ 2 } }$$

Evaluate

$$\frac{23\sqrt{14}}{14}\approx 6.147008564$$

Solution Steps

Rewrite the square root of the division \sqrt{\frac{2}{7}} as the division of square roots \frac{\sqrt{2}}{\sqrt{7}}.

$$\sqrt{14}+\frac{\sqrt{2}}{\sqrt{7}}+\sqrt{\frac{7}{2}}$$

Rationalize the denominator of \frac{\sqrt{2}}{\sqrt{7}} by multiplying numerator and denominator by \sqrt{7}.

$$\sqrt{14}+\frac{\sqrt{2}\sqrt{7}}{\left(\sqrt{7}\right)^{2}}+\sqrt{\frac{7}{2}}$$

The square of \sqrt{7} is 7.

$$\sqrt{14}+\frac{\sqrt{2}\sqrt{7}}{7}+\sqrt{\frac{7}{2}}$$

To multiply \sqrt{2} and \sqrt{7}, multiply the numbers under the square root.

$$\sqrt{14}+\frac{\sqrt{14}}{7}+\sqrt{\frac{7}{2}}$$

Combine \sqrt{14} and \frac{\sqrt{14}}{7} to get \frac{8}{7}\sqrt{14}.

$$\frac{8}{7}\sqrt{14}+\sqrt{\frac{7}{2}}$$

Rewrite the square root of the division \sqrt{\frac{7}{2}} as the division of square roots \frac{\sqrt{7}}{\sqrt{2}}.

$$\frac{8}{7}\sqrt{14}+\frac{\sqrt{7}}{\sqrt{2}}$$

Rationalize the denominator of \frac{\sqrt{7}}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{8}{7}\sqrt{14}+\frac{\sqrt{7}\sqrt{2}}{\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{8}{7}\sqrt{14}+\frac{\sqrt{7}\sqrt{2}}{2}$$

To multiply \sqrt{7} and \sqrt{2}, multiply the numbers under the square root.

$$\frac{8}{7}\sqrt{14}+\frac{\sqrt{14}}{2}$$

Combine \frac{8}{7}\sqrt{14} and \frac{\sqrt{14}}{2} to get \frac{23}{14}\sqrt{14}.

$$\frac{23}{14}\sqrt{14}$$