Solve sqrt{(1/3+1/6+1/9-1/12):(frac{1}{2}+frac{4}{5}-frac{2}{3})+(frac{7}{8}-frac{3}{4})+frac{1}{3}}+sqrt{frac{1}{5^4}:frac{1}{5^3}*frac{1}{5}}

$$\sqrt { ( \frac { 1 } { 3 } + \frac { 1 } { 6 } + \frac { 1 } { 9 } - \frac { 1 } { 12 } ) : ( \frac { 1 } { 2 } + \frac { 4 } { 5 } - \frac { 2 } { 3 } ) + ( \frac { 7 } { 8 } - \frac { 3 } { 4 } ) + \frac { 1 } { 3 } } + \sqrt { \frac { 1 } { 5 ^ { 4 } } : \frac { 1 } { 5 ^ { 3 } } \times \frac { 1 } { 5 } }$$

Evaluate

$$\frac{\sqrt{186}}{12}+\frac{1}{5}\approx 1.336515141$$

Short Solution Steps

Least common multiple of 3 and 6 is 6. Convert \frac{1}{3} and \frac{1}{6} to fractions with denominator 6.

$$\sqrt{\frac{\frac{2}{6}+\frac{1}{6}+\frac{1}{9}-\frac{1}{12}}{\frac{1}{2}+\frac{4}{5}-\frac{2}{3}}+\frac{7}{8}-\frac{3}{4}+\frac{1}{3}}+\sqrt{\frac{\frac{1}{5^{4}}}{\frac{1}{5^{3}}}\times \frac{1}{5}}$$

Since \frac{2}{6} and \frac{1}{6} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{\frac{2+1}{6}+\frac{1}{9}-\frac{1}{12}}{\frac{1}{2}+\frac{4}{5}-\frac{2}{3}}+\frac{7}{8}-\frac{3}{4}+\frac{1}{3}}+\sqrt{\frac{\frac{1}{5^{4}}}{\frac{1}{5^{3}}}\times \frac{1}{5}}$$

Add 2 and 1 to get 3.

$$\sqrt{\frac{\frac{3}{6}+\frac{1}{9}-\frac{1}{12}}{\frac{1}{2}+\frac{4}{5}-\frac{2}{3}}+\frac{7}{8}-\frac{3}{4}+\frac{1}{3}}+\sqrt{\frac{\frac{1}{5^{4}}}{\frac{1}{5^{3}}}\times \frac{1}{5}}$$

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

$$\sqrt{\frac{\frac{1}{2}+\frac{1}{9}-\frac{1}{12}}{\frac{1}{2}+\frac{4}{5}-\frac{2}{3}}+\frac{7}{8}-\frac{3}{4}+\frac{1}{3}}+\sqrt{\frac{\frac{1}{5^{4}}}{\frac{1}{5^{3}}}\times \frac{1}{5}}$$

Least common multiple of 2 and 9 is 18. Convert \frac{1}{2} and \frac{1}{9} to fractions with denominator 18.

$$\sqrt{\frac{\frac{9}{18}+\frac{2}{18}-\frac{1}{12}}{\frac{1}{2}+\frac{4}{5}-\frac{2}{3}}+\frac{7}{8}-\frac{3}{4}+\frac{1}{3}}+\sqrt{\frac{\frac{1}{5^{4}}}{\frac{1}{5^{3}}}\times \frac{1}{5}}$$

Since \frac{9}{18} and \frac{2}{18} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{\frac{9+2}{18}-\frac{1}{12}}{\frac{1}{2}+\frac{4}{5}-\frac{2}{3}}+\frac{7}{8}-\frac{3}{4}+\frac{1}{3}}+\sqrt{\frac{\frac{1}{5^{4}}}{\frac{1}{5^{3}}}\times \frac{1}{5}}$$

Add 9 and 2 to get 11.

$$\sqrt{\frac{\frac{11}{18}-\frac{1}{12}}{\frac{1}{2}+\frac{4}{5}-\frac{2}{3}}+\frac{7}{8}-\frac{3}{4}+\frac{1}{3}}+\sqrt{\frac{\frac{1}{5^{4}}}{\frac{1}{5^{3}}}\times \frac{1}{5}}$$

Least common multiple of 18 and 12 is 36. Convert \frac{11}{18} and \frac{1}{12} to fractions with denominator 36.

$$\sqrt{\frac{\frac{22}{36}-\frac{3}{36}}{\frac{1}{2}+\frac{4}{5}-\frac{2}{3}}+\frac{7}{8}-\frac{3}{4}+\frac{1}{3}}+\sqrt{\frac{\frac{1}{5^{4}}}{\frac{1}{5^{3}}}\times \frac{1}{5}}$$

Since \frac{22}{36} and \frac{3}{36} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{\frac{22-3}{36}}{\frac{1}{2}+\frac{4}{5}-\frac{2}{3}}+\frac{7}{8}-\frac{3}{4}+\frac{1}{3}}+\sqrt{\frac{\frac{1}{5^{4}}}{\frac{1}{5^{3}}}\times \frac{1}{5}}$$

Subtract 3 from 22 to get 19.

$$\sqrt{\frac{\frac{19}{36}}{\frac{1}{2}+\frac{4}{5}-\frac{2}{3}}+\frac{7}{8}-\frac{3}{4}+\frac{1}{3}}+\sqrt{\frac{\frac{1}{5^{4}}}{\frac{1}{5^{3}}}\times \frac{1}{5}}$$

Least common multiple of 2 and 5 is 10. Convert \frac{1}{2} and \frac{4}{5} to fractions with denominator 10.

$$\sqrt{\frac{\frac{19}{36}}{\frac{5}{10}+\frac{8}{10}-\frac{2}{3}}+\frac{7}{8}-\frac{3}{4}+\frac{1}{3}}+\sqrt{\frac{\frac{1}{5^{4}}}{\frac{1}{5^{3}}}\times \frac{1}{5}}$$

Since \frac{5}{10} and \frac{8}{10} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{\frac{19}{36}}{\frac{5+8}{10}-\frac{2}{3}}+\frac{7}{8}-\frac{3}{4}+\frac{1}{3}}+\sqrt{\frac{\frac{1}{5^{4}}}{\frac{1}{5^{3}}}\times \frac{1}{5}}$$

Add 5 and 8 to get 13.

$$\sqrt{\frac{\frac{19}{36}}{\frac{13}{10}-\frac{2}{3}}+\frac{7}{8}-\frac{3}{4}+\frac{1}{3}}+\sqrt{\frac{\frac{1}{5^{4}}}{\frac{1}{5^{3}}}\times \frac{1}{5}}$$

Least common multiple of 10 and 3 is 30. Convert \frac{13}{10} and \frac{2}{3} to fractions with denominator 30.

$$\sqrt{\frac{\frac{19}{36}}{\frac{39}{30}-\frac{20}{30}}+\frac{7}{8}-\frac{3}{4}+\frac{1}{3}}+\sqrt{\frac{\frac{1}{5^{4}}}{\frac{1}{5^{3}}}\times \frac{1}{5}}$$

Since \frac{39}{30} and \frac{20}{30} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{\frac{19}{36}}{\frac{39-20}{30}}+\frac{7}{8}-\frac{3}{4}+\frac{1}{3}}+\sqrt{\frac{\frac{1}{5^{4}}}{\frac{1}{5^{3}}}\times \frac{1}{5}}$$

Subtract 20 from 39 to get 19.

$$\sqrt{\frac{\frac{19}{36}}{\frac{19}{30}}+\frac{7}{8}-\frac{3}{4}+\frac{1}{3}}+\sqrt{\frac{\frac{1}{5^{4}}}{\frac{1}{5^{3}}}\times \frac{1}{5}}$$

Divide \frac{19}{36} by \frac{19}{30} by multiplying \frac{19}{36} by the reciprocal of \frac{19}{30}.

$$\sqrt{\frac{19}{36}\times \frac{30}{19}+\frac{7}{8}-\frac{3}{4}+\frac{1}{3}}+\sqrt{\frac{\frac{1}{5^{4}}}{\frac{1}{5^{3}}}\times \frac{1}{5}}$$

Multiply \frac{19}{36} times \frac{30}{19} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{19\times 30}{36\times 19}+\frac{7}{8}-\frac{3}{4}+\frac{1}{3}}+\sqrt{\frac{\frac{1}{5^{4}}}{\frac{1}{5^{3}}}\times \frac{1}{5}}$$

Cancel out 19 in both numerator and denominator.

$$\sqrt{\frac{30}{36}+\frac{7}{8}-\frac{3}{4}+\frac{1}{3}}+\sqrt{\frac{\frac{1}{5^{4}}}{\frac{1}{5^{3}}}\times \frac{1}{5}}$$

Reduce the fraction \frac{30}{36} to lowest terms by extracting and canceling out 6.

$$\sqrt{\frac{5}{6}+\frac{7}{8}-\frac{3}{4}+\frac{1}{3}}+\sqrt{\frac{\frac{1}{5^{4}}}{\frac{1}{5^{3}}}\times \frac{1}{5}}$$

Least common multiple of 6 and 8 is 24. Convert \frac{5}{6} and \frac{7}{8} to fractions with denominator 24.

$$\sqrt{\frac{20}{24}+\frac{21}{24}-\frac{3}{4}+\frac{1}{3}}+\sqrt{\frac{\frac{1}{5^{4}}}{\frac{1}{5^{3}}}\times \frac{1}{5}}$$

Since \frac{20}{24} and \frac{21}{24} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{20+21}{24}-\frac{3}{4}+\frac{1}{3}}+\sqrt{\frac{\frac{1}{5^{4}}}{\frac{1}{5^{3}}}\times \frac{1}{5}}$$

Add 20 and 21 to get 41.

$$\sqrt{\frac{41}{24}-\frac{3}{4}+\frac{1}{3}}+\sqrt{\frac{\frac{1}{5^{4}}}{\frac{1}{5^{3}}}\times \frac{1}{5}}$$

Least common multiple of 24 and 4 is 24. Convert \frac{41}{24} and \frac{3}{4} to fractions with denominator 24.

$$\sqrt{\frac{41}{24}-\frac{18}{24}+\frac{1}{3}}+\sqrt{\frac{\frac{1}{5^{4}}}{\frac{1}{5^{3}}}\times \frac{1}{5}}$$

Since \frac{41}{24} and \frac{18}{24} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{41-18}{24}+\frac{1}{3}}+\sqrt{\frac{\frac{1}{5^{4}}}{\frac{1}{5^{3}}}\times \frac{1}{5}}$$

Subtract 18 from 41 to get 23.

$$\sqrt{\frac{23}{24}+\frac{1}{3}}+\sqrt{\frac{\frac{1}{5^{4}}}{\frac{1}{5^{3}}}\times \frac{1}{5}}$$

Least common multiple of 24 and 3 is 24. Convert \frac{23}{24} and \frac{1}{3} to fractions with denominator 24.

$$\sqrt{\frac{23}{24}+\frac{8}{24}}+\sqrt{\frac{\frac{1}{5^{4}}}{\frac{1}{5^{3}}}\times \frac{1}{5}}$$

Since \frac{23}{24} and \frac{8}{24} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{23+8}{24}}+\sqrt{\frac{\frac{1}{5^{4}}}{\frac{1}{5^{3}}}\times \frac{1}{5}}$$

Add 23 and 8 to get 31.

$$\sqrt{\frac{31}{24}}+\sqrt{\frac{\frac{1}{5^{4}}}{\frac{1}{5^{3}}}\times \frac{1}{5}}$$

Rewrite the square root of the division \sqrt{\frac{31}{24}} as the division of square roots \frac{\sqrt{31}}{\sqrt{24}}.

$$\frac{\sqrt{31}}{\sqrt{24}}+\sqrt{\frac{\frac{1}{5^{4}}}{\frac{1}{5^{3}}}\times \frac{1}{5}}$$

Factor 24=2^{2}\times 6. Rewrite the square root of the product \sqrt{2^{2}\times 6} as the product of square roots \sqrt{2^{2}}\sqrt{6}. Take the square root of 2^{2}.

$$\frac{\sqrt{31}}{2\sqrt{6}}+\sqrt{\frac{\frac{1}{5^{4}}}{\frac{1}{5^{3}}}\times \frac{1}{5}}$$

Rationalize the denominator of \frac{\sqrt{31}}{2\sqrt{6}} by multiplying numerator and denominator by \sqrt{6}.

$$\frac{\sqrt{31}\sqrt{6}}{2\left(\sqrt{6}\right)^{2}}+\sqrt{\frac{\frac{1}{5^{4}}}{\frac{1}{5^{3}}}\times \frac{1}{5}}$$

The square of \sqrt{6} is 6.

$$\frac{\sqrt{31}\sqrt{6}}{2\times 6}+\sqrt{\frac{\frac{1}{5^{4}}}{\frac{1}{5^{3}}}\times \frac{1}{5}}$$

To multiply \sqrt{31} and \sqrt{6}, multiply the numbers under the square root.

$$\frac{\sqrt{186}}{2\times 6}+\sqrt{\frac{\frac{1}{5^{4}}}{\frac{1}{5^{3}}}\times \frac{1}{5}}$$

Multiply 2 and 6 to get 12.

$$\frac{\sqrt{186}}{12}+\sqrt{\frac{\frac{1}{5^{4}}}{\frac{1}{5^{3}}}\times \frac{1}{5}}$$

Divide \frac{1}{5^{4}} by \frac{1}{5^{3}} by multiplying \frac{1}{5^{4}} by the reciprocal of \frac{1}{5^{3}}.

$$\frac{\sqrt{186}}{12}+\sqrt{\frac{5^{3}}{5^{4}}\times \frac{1}{5}}$$

Cancel out 5^{3} in both numerator and denominator.

$$\frac{\sqrt{186}}{12}+\sqrt{\frac{1}{5}\times \frac{1}{5}}$$

Multiply \frac{1}{5} times \frac{1}{5} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\sqrt{186}}{12}+\sqrt{\frac{1\times 1}{5\times 5}}$$

Do the multiplications in the fraction \frac{1\times 1}{5\times 5}.

$$\frac{\sqrt{186}}{12}+\sqrt{\frac{1}{25}}$$

Rewrite the square root of the division \frac{1}{25} as the division of square roots \frac{\sqrt{1}}{\sqrt{25}}. Take the square root of both numerator and denominator.

$$\frac{\sqrt{186}}{12}+\frac{1}{5}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 12 and 5 is 60. Multiply \frac{\sqrt{186}}{12} times \frac{5}{5}. Multiply \frac{1}{5} times \frac{12}{12}.

$$\frac{5\sqrt{186}}{60}+\frac{12}{60}$$

Since \frac{5\sqrt{186}}{60} and \frac{12}{60} have the same denominator, add them by adding their numerators.

$$\frac{5\sqrt{186}+12}{60}$$