Solve sqrt{(2div23+45div89)}

$$\sqrt{ (2 \div 23+45 \div 89) }$$

Evaluate

$$\frac{\sqrt{2483011}}{2047}\approx 0.769788607$$

Solution Steps

Least common multiple of 23 and 89 is 2047. Convert \frac{2}{23} and \frac{45}{89} to fractions with denominator 2047.

$$\sqrt{\frac{178}{2047}+\frac{1035}{2047}}$$

Since \frac{178}{2047} and \frac{1035}{2047} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{178+1035}{2047}}$$

Add 178 and 1035 to get 1213.

$$\sqrt{\frac{1213}{2047}}$$

Rewrite the square root of the division \sqrt{\frac{1213}{2047}} as the division of square roots \frac{\sqrt{1213}}{\sqrt{2047}}.

$$\frac{\sqrt{1213}}{\sqrt{2047}}$$

Rationalize the denominator of \frac{\sqrt{1213}}{\sqrt{2047}} by multiplying numerator and denominator by \sqrt{2047}.

$$\frac{\sqrt{1213}\sqrt{2047}}{\left(\sqrt{2047}\right)^{2}}$$

The square of \sqrt{2047} is 2047.

$$\frac{\sqrt{1213}\sqrt{2047}}{2047}$$

To multiply \sqrt{1213} and \sqrt{2047}, multiply the numbers under the square root.

$$\frac{\sqrt{2483011}}{2047}$$