Solve sqrt{8/9}+sqrt{18/36}

$$\sqrt { \frac { 8 } { 9 } } + \sqrt { \frac { 18 } { 36 } }$$

Evaluate

$$\frac{7\sqrt{2}}{6}\approx 1.649915823$$

Solution Steps

Rewrite the square root of the division \sqrt{\frac{8}{9}} as the division of square roots \frac{\sqrt{8}}{\sqrt{9}}.

$$\frac{\sqrt{8}}{\sqrt{9}}+\sqrt{\frac{18}{36}}$$

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

$$\frac{2\sqrt{2}}{\sqrt{9}}+\sqrt{\frac{18}{36}}$$

Calculate the square root of 9 and get 3.

$$\frac{2\sqrt{2}}{3}+\sqrt{\frac{18}{36}}$$

Reduce the fraction \frac{18}{36} to lowest terms by extracting and canceling out 18.

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

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

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

Calculate the square root of 1 and get 1.

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

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

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

The square of \sqrt{2} is 2.

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

Combine \frac{2\sqrt{2}}{3} and \frac{\sqrt{2}}{2} to get \frac{7}{6}\sqrt{2}.

$$\frac{7}{6}\sqrt{2}$$