Solve sqrt{[(1/5+3/4*8/9):13/5]*(frac{5}{6}+frac{3}{2}-frac{1}{12})+frac{1}{2}}

$$\sqrt { [ ( \frac { 1 } { 5 } + \frac { 3 } { 4 } \cdot \frac { 8 } { 9 } ) : \frac { 13 } { 5 } ] \cdot ( \frac { 5 } { 6 } + \frac { 3 } { 2 } - \frac { 1 } { 12 } ) + \frac { 1 } { 2 } }$$

Evaluate

$$\frac{\sqrt{5}}{2}\approx 1.118033989$$

Solution Steps

Multiply \frac{3}{4} times \frac{8}{9} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{\frac{1}{5}+\frac{3\times 8}{4\times 9}}{\frac{13}{5}}\left(\frac{5}{6}+\frac{3}{2}-\frac{1}{12}\right)+\frac{1}{2}}$$

Do the multiplications in the fraction \frac{3\times 8}{4\times 9}.

$$\sqrt{\frac{\frac{1}{5}+\frac{24}{36}}{\frac{13}{5}}\left(\frac{5}{6}+\frac{3}{2}-\frac{1}{12}\right)+\frac{1}{2}}$$

Reduce the fraction \frac{24}{36} to lowest terms by extracting and canceling out 12.

$$\sqrt{\frac{\frac{1}{5}+\frac{2}{3}}{\frac{13}{5}}\left(\frac{5}{6}+\frac{3}{2}-\frac{1}{12}\right)+\frac{1}{2}}$$

Least common multiple of 5 and 3 is 15. Convert \frac{1}{5} and \frac{2}{3} to fractions with denominator 15.

$$\sqrt{\frac{\frac{3}{15}+\frac{10}{15}}{\frac{13}{5}}\left(\frac{5}{6}+\frac{3}{2}-\frac{1}{12}\right)+\frac{1}{2}}$$

Since \frac{3}{15} and \frac{10}{15} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{\frac{3+10}{15}}{\frac{13}{5}}\left(\frac{5}{6}+\frac{3}{2}-\frac{1}{12}\right)+\frac{1}{2}}$$

Add 3 and 10 to get 13.

$$\sqrt{\frac{\frac{13}{15}}{\frac{13}{5}}\left(\frac{5}{6}+\frac{3}{2}-\frac{1}{12}\right)+\frac{1}{2}}$$

Divide \frac{13}{15} by \frac{13}{5} by multiplying \frac{13}{15} by the reciprocal of \frac{13}{5}.

$$\sqrt{\frac{13}{15}\times \frac{5}{13}\left(\frac{5}{6}+\frac{3}{2}-\frac{1}{12}\right)+\frac{1}{2}}$$

Multiply \frac{13}{15} times \frac{5}{13} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{13\times 5}{15\times 13}\left(\frac{5}{6}+\frac{3}{2}-\frac{1}{12}\right)+\frac{1}{2}}$$

Cancel out 13 in both numerator and denominator.

$$\sqrt{\frac{5}{15}\left(\frac{5}{6}+\frac{3}{2}-\frac{1}{12}\right)+\frac{1}{2}}$$

Reduce the fraction \frac{5}{15} to lowest terms by extracting and canceling out 5.

$$\sqrt{\frac{1}{3}\left(\frac{5}{6}+\frac{3}{2}-\frac{1}{12}\right)+\frac{1}{2}}$$

Least common multiple of 6 and 2 is 6. Convert \frac{5}{6} and \frac{3}{2} to fractions with denominator 6.

$$\sqrt{\frac{1}{3}\left(\frac{5}{6}+\frac{9}{6}-\frac{1}{12}\right)+\frac{1}{2}}$$

Since \frac{5}{6} and \frac{9}{6} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{1}{3}\left(\frac{5+9}{6}-\frac{1}{12}\right)+\frac{1}{2}}$$

Add 5 and 9 to get 14.

$$\sqrt{\frac{1}{3}\left(\frac{14}{6}-\frac{1}{12}\right)+\frac{1}{2}}$$

Reduce the fraction \frac{14}{6} to lowest terms by extracting and canceling out 2.

$$\sqrt{\frac{1}{3}\left(\frac{7}{3}-\frac{1}{12}\right)+\frac{1}{2}}$$

Least common multiple of 3 and 12 is 12. Convert \frac{7}{3} and \frac{1}{12} to fractions with denominator 12.

$$\sqrt{\frac{1}{3}\left(\frac{28}{12}-\frac{1}{12}\right)+\frac{1}{2}}$$

Since \frac{28}{12} and \frac{1}{12} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{1}{3}\times \frac{28-1}{12}+\frac{1}{2}}$$

Subtract 1 from 28 to get 27.

$$\sqrt{\frac{1}{3}\times \frac{27}{12}+\frac{1}{2}}$$

Reduce the fraction \frac{27}{12} to lowest terms by extracting and canceling out 3.

$$\sqrt{\frac{1}{3}\times \frac{9}{4}+\frac{1}{2}}$$

Multiply \frac{1}{3} times \frac{9}{4} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{1\times 9}{3\times 4}+\frac{1}{2}}$$

Do the multiplications in the fraction \frac{1\times 9}{3\times 4}.

$$\sqrt{\frac{9}{12}+\frac{1}{2}}$$

Reduce the fraction \frac{9}{12} to lowest terms by extracting and canceling out 3.

$$\sqrt{\frac{3}{4}+\frac{1}{2}}$$

Least common multiple of 4 and 2 is 4. Convert \frac{3}{4} and \frac{1}{2} to fractions with denominator 4.

$$\sqrt{\frac{3}{4}+\frac{2}{4}}$$

Since \frac{3}{4} and \frac{2}{4} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{3+2}{4}}$$

Add 3 and 2 to get 5.

$$\sqrt{\frac{5}{4}}$$

Rewrite the square root of the division \sqrt{\frac{5}{4}} as the division of square roots \frac{\sqrt{5}}{\sqrt{4}}.

$$\frac{\sqrt{5}}{\sqrt{4}}$$

Calculate the square root of 4 and get 2.

$$\frac{\sqrt{5}}{2}$$