Solve sqrt{23/180+8/9}:13/20

$$\sqrt { \frac { 23 } { 180 } + \frac { 8 } { 9 } } : \frac { 13 } { 20 }$$

Evaluate

$$\frac{2\sqrt{915}}{39}\approx 1.551229073$$

Solution Steps

Least common multiple of 180 and 9 is 180. Convert \frac{23}{180} and \frac{8}{9} to fractions with denominator 180.

$$\frac{\sqrt{\frac{23}{180}+\frac{160}{180}}}{\frac{13}{20}}$$

Since \frac{23}{180} and \frac{160}{180} have the same denominator, add them by adding their numerators.

$$\frac{\sqrt{\frac{23+160}{180}}}{\frac{13}{20}}$$

Add 23 and 160 to get 183.

$$\frac{\sqrt{\frac{183}{180}}}{\frac{13}{20}}$$

Reduce the fraction \frac{183}{180} to lowest terms by extracting and canceling out 3.

$$\frac{\sqrt{\frac{61}{60}}}{\frac{13}{20}}$$

Rewrite the square root of the division \sqrt{\frac{61}{60}} as the division of square roots \frac{\sqrt{61}}{\sqrt{60}}.

$$\frac{\frac{\sqrt{61}}{\sqrt{60}}}{\frac{13}{20}}$$

Factor 60=2^{2}\times 15. Rewrite the square root of the product \sqrt{2^{2}\times 15} as the product of square roots \sqrt{2^{2}}\sqrt{15}. Take the square root of 2^{2}.

$$\frac{\frac{\sqrt{61}}{2\sqrt{15}}}{\frac{13}{20}}$$

Rationalize the denominator of \frac{\sqrt{61}}{2\sqrt{15}} by multiplying numerator and denominator by \sqrt{15}.

$$\frac{\frac{\sqrt{61}\sqrt{15}}{2\left(\sqrt{15}\right)^{2}}}{\frac{13}{20}}$$

The square of \sqrt{15} is 15.

$$\frac{\frac{\sqrt{61}\sqrt{15}}{2\times 15}}{\frac{13}{20}}$$

To multiply \sqrt{61} and \sqrt{15}, multiply the numbers under the square root.

$$\frac{\frac{\sqrt{915}}{2\times 15}}{\frac{13}{20}}$$

Multiply 2 and 15 to get 30.

$$\frac{\frac{\sqrt{915}}{30}}{\frac{13}{20}}$$

Divide \frac{\sqrt{915}}{30} by \frac{13}{20} by multiplying \frac{\sqrt{915}}{30} by the reciprocal of \frac{13}{20}.

$$\frac{\sqrt{915}\times 20}{30\times 13}$$

Cancel out 10 in both numerator and denominator.

$$\frac{2\sqrt{915}}{3\times 13}$$

Multiply 3 and 13 to get 39.

$$\frac{2\sqrt{915}}{39}$$