Solve sqrt{18/27+19/27}

$$\sqrt{ \frac{ 18 }{ 27 } + \frac{ 19 }{ 27 } }$$

Evaluate

$$\frac{\sqrt{111}}{9}\approx 1.170628195$$

Solution Steps

Since \frac{18}{27} and \frac{19}{27} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{18+19}{27}}$$

Add 18 and 19 to get 37.

$$\sqrt{\frac{37}{27}}$$

Rewrite the square root of the division \sqrt{\frac{37}{27}} as the division of square roots \frac{\sqrt{37}}{\sqrt{27}}.

$$\frac{\sqrt{37}}{\sqrt{27}}$$

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}.

$$\frac{\sqrt{37}}{3\sqrt{3}}$$

Rationalize the denominator of \frac{\sqrt{37}}{3\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{\sqrt{37}\sqrt{3}}{3\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{\sqrt{37}\sqrt{3}}{3\times 3}$$

To multiply \sqrt{37} and \sqrt{3}, multiply the numbers under the square root.

$$\frac{\sqrt{111}}{3\times 3}$$

Multiply 3 and 3 to get 9.

$$\frac{\sqrt{111}}{9}$$