Solve sqrt{(5/12+3/6-1/2)*(2/3+frac{1}{8})-frac{1}{3}+(1+frac{5}{9})}

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

Evaluate

$$\frac{\sqrt{894}}{24}\approx 1.245826366$$

Solution Steps

Reduce the fraction \frac{3}{6} to lowest terms by extracting and canceling out 3.

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

Least common multiple of 12 and 2 is 12. Convert \frac{5}{12} and \frac{1}{2} to fractions with denominator 12.

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

Since \frac{5}{12} and \frac{6}{12} have the same denominator, add them by adding their numerators.

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

Add 5 and 6 to get 11.

$$\sqrt{\left(\frac{11}{12}-\frac{1}{2}\right)\left(\frac{2}{3}+\frac{1}{8}\right)-\frac{1}{3}+1+\frac{5}{9}}$$

Least common multiple of 12 and 2 is 12. Convert \frac{11}{12} and \frac{1}{2} to fractions with denominator 12.

$$\sqrt{\left(\frac{11}{12}-\frac{6}{12}\right)\left(\frac{2}{3}+\frac{1}{8}\right)-\frac{1}{3}+1+\frac{5}{9}}$$

Since \frac{11}{12} and \frac{6}{12} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{11-6}{12}\left(\frac{2}{3}+\frac{1}{8}\right)-\frac{1}{3}+1+\frac{5}{9}}$$

Subtract 6 from 11 to get 5.

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

Least common multiple of 3 and 8 is 24. Convert \frac{2}{3} and \frac{1}{8} to fractions with denominator 24.

$$\sqrt{\frac{5}{12}\left(\frac{16}{24}+\frac{3}{24}\right)-\frac{1}{3}+1+\frac{5}{9}}$$

Since \frac{16}{24} and \frac{3}{24} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{5}{12}\times \frac{16+3}{24}-\frac{1}{3}+1+\frac{5}{9}}$$

Add 16 and 3 to get 19.

$$\sqrt{\frac{5}{12}\times \frac{19}{24}-\frac{1}{3}+1+\frac{5}{9}}$$

Multiply \frac{5}{12} times \frac{19}{24} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{5\times 19}{12\times 24}-\frac{1}{3}+1+\frac{5}{9}}$$

Do the multiplications in the fraction \frac{5\times 19}{12\times 24}.

$$\sqrt{\frac{95}{288}-\frac{1}{3}+1+\frac{5}{9}}$$

Least common multiple of 288 and 3 is 288. Convert \frac{95}{288} and \frac{1}{3} to fractions with denominator 288.

$$\sqrt{\frac{95}{288}-\frac{96}{288}+1+\frac{5}{9}}$$

Since \frac{95}{288} and \frac{96}{288} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{95-96}{288}+1+\frac{5}{9}}$$

Subtract 96 from 95 to get -1.

$$\sqrt{-\frac{1}{288}+1+\frac{5}{9}}$$

Convert 1 to fraction \frac{288}{288}.

$$\sqrt{-\frac{1}{288}+\frac{288}{288}+\frac{5}{9}}$$

Since -\frac{1}{288} and \frac{288}{288} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{-1+288}{288}+\frac{5}{9}}$$

Add -1 and 288 to get 287.

$$\sqrt{\frac{287}{288}+\frac{5}{9}}$$

Least common multiple of 288 and 9 is 288. Convert \frac{287}{288} and \frac{5}{9} to fractions with denominator 288.

$$\sqrt{\frac{287}{288}+\frac{160}{288}}$$

Since \frac{287}{288} and \frac{160}{288} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{287+160}{288}}$$

Add 287 and 160 to get 447.

$$\sqrt{\frac{447}{288}}$$

Reduce the fraction \frac{447}{288} to lowest terms by extracting and canceling out 3.

$$\sqrt{\frac{149}{96}}$$

Rewrite the square root of the division \sqrt{\frac{149}{96}} as the division of square roots \frac{\sqrt{149}}{\sqrt{96}}.

$$\frac{\sqrt{149}}{\sqrt{96}}$$

Factor 96=4^{2}\times 6. Rewrite the square root of the product \sqrt{4^{2}\times 6} as the product of square roots \sqrt{4^{2}}\sqrt{6}. Take the square root of 4^{2}.

$$\frac{\sqrt{149}}{4\sqrt{6}}$$

Rationalize the denominator of \frac{\sqrt{149}}{4\sqrt{6}} by multiplying numerator and denominator by \sqrt{6}.

$$\frac{\sqrt{149}\sqrt{6}}{4\left(\sqrt{6}\right)^{2}}$$

The square of \sqrt{6} is 6.

$$\frac{\sqrt{149}\sqrt{6}}{4\times 6}$$

To multiply \sqrt{149} and \sqrt{6}, multiply the numbers under the square root.

$$\frac{\sqrt{894}}{4\times 6}$$

Multiply 4 and 6 to get 24.

$$\frac{\sqrt{894}}{24}$$