Solve 2+sqrt{(1-1/5):{[3/2-1/6:(4/5*frac{10}{3}+2)]+(10-frac{5}{8}:frac{25}{24}+frac{3}{5})*frac{1}{4}}*3}

$$2 + \sqrt { ( 1 - \frac { 1 } { 5 } ) : \{ [ \frac { 3 } { 2 } - \frac { 1 } { 6 } : ( \frac { 4 } { 5 } \times \frac { 10 } { 3 } + 2 ) ] + ( 10 - \frac { 5 } { 8 } : \frac { 25 } { 24 } + \frac { 3 } { 5 } ) \times \frac { 1 } { 4 } \} \times 3 }$$

Evaluate

$$\frac{4\sqrt{1295}}{185}+2\approx 2.77807802$$

Short Solution Steps

Convert 1 to fraction \frac{5}{5}.

$$2+\sqrt{\frac{\frac{5}{5}-\frac{1}{5}}{\frac{3}{2}-\frac{\frac{1}{6}}{\frac{4}{5}\times \frac{10}{3}+2}+\left(10-\frac{\frac{5}{8}}{\frac{25}{24}}+\frac{3}{5}\right)\times \frac{1}{4}}\times 3}$$

Since \frac{5}{5} and \frac{1}{5} have the same denominator, subtract them by subtracting their numerators.

$$2+\sqrt{\frac{\frac{5-1}{5}}{\frac{3}{2}-\frac{\frac{1}{6}}{\frac{4}{5}\times \frac{10}{3}+2}+\left(10-\frac{\frac{5}{8}}{\frac{25}{24}}+\frac{3}{5}\right)\times \frac{1}{4}}\times 3}$$

Subtract 1 from 5 to get 4.

$$2+\sqrt{\frac{\frac{4}{5}}{\frac{3}{2}-\frac{\frac{1}{6}}{\frac{4}{5}\times \frac{10}{3}+2}+\left(10-\frac{\frac{5}{8}}{\frac{25}{24}}+\frac{3}{5}\right)\times \frac{1}{4}}\times 3}$$

Multiply \frac{4}{5} times \frac{10}{3} by multiplying numerator times numerator and denominator times denominator.

$$2+\sqrt{\frac{\frac{4}{5}}{\frac{3}{2}-\frac{\frac{1}{6}}{\frac{4\times 10}{5\times 3}+2}+\left(10-\frac{\frac{5}{8}}{\frac{25}{24}}+\frac{3}{5}\right)\times \frac{1}{4}}\times 3}$$

Do the multiplications in the fraction \frac{4\times 10}{5\times 3}.

$$2+\sqrt{\frac{\frac{4}{5}}{\frac{3}{2}-\frac{\frac{1}{6}}{\frac{40}{15}+2}+\left(10-\frac{\frac{5}{8}}{\frac{25}{24}}+\frac{3}{5}\right)\times \frac{1}{4}}\times 3}$$

Reduce the fraction \frac{40}{15} to lowest terms by extracting and canceling out 5.

$$2+\sqrt{\frac{\frac{4}{5}}{\frac{3}{2}-\frac{\frac{1}{6}}{\frac{8}{3}+2}+\left(10-\frac{\frac{5}{8}}{\frac{25}{24}}+\frac{3}{5}\right)\times \frac{1}{4}}\times 3}$$

Convert 2 to fraction \frac{6}{3}.

$$2+\sqrt{\frac{\frac{4}{5}}{\frac{3}{2}-\frac{\frac{1}{6}}{\frac{8}{3}+\frac{6}{3}}+\left(10-\frac{\frac{5}{8}}{\frac{25}{24}}+\frac{3}{5}\right)\times \frac{1}{4}}\times 3}$$

Since \frac{8}{3} and \frac{6}{3} have the same denominator, add them by adding their numerators.

$$2+\sqrt{\frac{\frac{4}{5}}{\frac{3}{2}-\frac{\frac{1}{6}}{\frac{8+6}{3}}+\left(10-\frac{\frac{5}{8}}{\frac{25}{24}}+\frac{3}{5}\right)\times \frac{1}{4}}\times 3}$$

Add 8 and 6 to get 14.

$$2+\sqrt{\frac{\frac{4}{5}}{\frac{3}{2}-\frac{\frac{1}{6}}{\frac{14}{3}}+\left(10-\frac{\frac{5}{8}}{\frac{25}{24}}+\frac{3}{5}\right)\times \frac{1}{4}}\times 3}$$

Divide \frac{1}{6} by \frac{14}{3} by multiplying \frac{1}{6} by the reciprocal of \frac{14}{3}.

$$2+\sqrt{\frac{\frac{4}{5}}{\frac{3}{2}-\frac{1}{6}\times \frac{3}{14}+\left(10-\frac{\frac{5}{8}}{\frac{25}{24}}+\frac{3}{5}\right)\times \frac{1}{4}}\times 3}$$

Multiply \frac{1}{6} times \frac{3}{14} by multiplying numerator times numerator and denominator times denominator.

$$2+\sqrt{\frac{\frac{4}{5}}{\frac{3}{2}-\frac{1\times 3}{6\times 14}+\left(10-\frac{\frac{5}{8}}{\frac{25}{24}}+\frac{3}{5}\right)\times \frac{1}{4}}\times 3}$$

Do the multiplications in the fraction \frac{1\times 3}{6\times 14}.

$$2+\sqrt{\frac{\frac{4}{5}}{\frac{3}{2}-\frac{3}{84}+\left(10-\frac{\frac{5}{8}}{\frac{25}{24}}+\frac{3}{5}\right)\times \frac{1}{4}}\times 3}$$

Reduce the fraction \frac{3}{84} to lowest terms by extracting and canceling out 3.

$$2+\sqrt{\frac{\frac{4}{5}}{\frac{3}{2}-\frac{1}{28}+\left(10-\frac{\frac{5}{8}}{\frac{25}{24}}+\frac{3}{5}\right)\times \frac{1}{4}}\times 3}$$

Least common multiple of 2 and 28 is 28. Convert \frac{3}{2} and \frac{1}{28} to fractions with denominator 28.

$$2+\sqrt{\frac{\frac{4}{5}}{\frac{42}{28}-\frac{1}{28}+\left(10-\frac{\frac{5}{8}}{\frac{25}{24}}+\frac{3}{5}\right)\times \frac{1}{4}}\times 3}$$

Since \frac{42}{28} and \frac{1}{28} have the same denominator, subtract them by subtracting their numerators.

$$2+\sqrt{\frac{\frac{4}{5}}{\frac{42-1}{28}+\left(10-\frac{\frac{5}{8}}{\frac{25}{24}}+\frac{3}{5}\right)\times \frac{1}{4}}\times 3}$$

Subtract 1 from 42 to get 41.

$$2+\sqrt{\frac{\frac{4}{5}}{\frac{41}{28}+\left(10-\frac{\frac{5}{8}}{\frac{25}{24}}+\frac{3}{5}\right)\times \frac{1}{4}}\times 3}$$

Divide \frac{5}{8} by \frac{25}{24} by multiplying \frac{5}{8} by the reciprocal of \frac{25}{24}.

$$2+\sqrt{\frac{\frac{4}{5}}{\frac{41}{28}+\left(10-\frac{5}{8}\times \frac{24}{25}+\frac{3}{5}\right)\times \frac{1}{4}}\times 3}$$

Multiply \frac{5}{8} times \frac{24}{25} by multiplying numerator times numerator and denominator times denominator.

$$2+\sqrt{\frac{\frac{4}{5}}{\frac{41}{28}+\left(10-\frac{5\times 24}{8\times 25}+\frac{3}{5}\right)\times \frac{1}{4}}\times 3}$$

Do the multiplications in the fraction \frac{5\times 24}{8\times 25}.

$$2+\sqrt{\frac{\frac{4}{5}}{\frac{41}{28}+\left(10-\frac{120}{200}+\frac{3}{5}\right)\times \frac{1}{4}}\times 3}$$

Reduce the fraction \frac{120}{200} to lowest terms by extracting and canceling out 40.

$$2+\sqrt{\frac{\frac{4}{5}}{\frac{41}{28}+\left(10-\frac{3}{5}+\frac{3}{5}\right)\times \frac{1}{4}}\times 3}$$

Convert 10 to fraction \frac{50}{5}.

$$2+\sqrt{\frac{\frac{4}{5}}{\frac{41}{28}+\left(\frac{50}{5}-\frac{3}{5}+\frac{3}{5}\right)\times \frac{1}{4}}\times 3}$$

Since \frac{50}{5} and \frac{3}{5} have the same denominator, subtract them by subtracting their numerators.

$$2+\sqrt{\frac{\frac{4}{5}}{\frac{41}{28}+\left(\frac{50-3}{5}+\frac{3}{5}\right)\times \frac{1}{4}}\times 3}$$

Subtract 3 from 50 to get 47.

$$2+\sqrt{\frac{\frac{4}{5}}{\frac{41}{28}+\left(\frac{47}{5}+\frac{3}{5}\right)\times \frac{1}{4}}\times 3}$$

Since \frac{47}{5} and \frac{3}{5} have the same denominator, add them by adding their numerators.

$$2+\sqrt{\frac{\frac{4}{5}}{\frac{41}{28}+\frac{47+3}{5}\times \frac{1}{4}}\times 3}$$

Add 47 and 3 to get 50.

$$2+\sqrt{\frac{\frac{4}{5}}{\frac{41}{28}+\frac{50}{5}\times \frac{1}{4}}\times 3}$$

Divide 50 by 5 to get 10.

$$2+\sqrt{\frac{\frac{4}{5}}{\frac{41}{28}+10\times \frac{1}{4}}\times 3}$$

Multiply 10 and \frac{1}{4} to get \frac{10}{4}.

$$2+\sqrt{\frac{\frac{4}{5}}{\frac{41}{28}+\frac{10}{4}}\times 3}$$

Reduce the fraction \frac{10}{4} to lowest terms by extracting and canceling out 2.

$$2+\sqrt{\frac{\frac{4}{5}}{\frac{41}{28}+\frac{5}{2}}\times 3}$$

Least common multiple of 28 and 2 is 28. Convert \frac{41}{28} and \frac{5}{2} to fractions with denominator 28.

$$2+\sqrt{\frac{\frac{4}{5}}{\frac{41}{28}+\frac{70}{28}}\times 3}$$

Since \frac{41}{28} and \frac{70}{28} have the same denominator, add them by adding their numerators.

$$2+\sqrt{\frac{\frac{4}{5}}{\frac{41+70}{28}}\times 3}$$

Add 41 and 70 to get 111.

$$2+\sqrt{\frac{\frac{4}{5}}{\frac{111}{28}}\times 3}$$

Divide \frac{4}{5} by \frac{111}{28} by multiplying \frac{4}{5} by the reciprocal of \frac{111}{28}.

$$2+\sqrt{\frac{4}{5}\times \frac{28}{111}\times 3}$$

Multiply \frac{4}{5} times \frac{28}{111} by multiplying numerator times numerator and denominator times denominator.

$$2+\sqrt{\frac{4\times 28}{5\times 111}\times 3}$$

Do the multiplications in the fraction \frac{4\times 28}{5\times 111}.

$$2+\sqrt{\frac{112}{555}\times 3}$$

Express \frac{112}{555}\times 3 as a single fraction.

$$2+\sqrt{\frac{112\times 3}{555}}$$

Multiply 112 and 3 to get 336.

$$2+\sqrt{\frac{336}{555}}$$

Reduce the fraction \frac{336}{555} to lowest terms by extracting and canceling out 3.

$$2+\sqrt{\frac{112}{185}}$$

Rewrite the square root of the division \sqrt{\frac{112}{185}} as the division of square roots \frac{\sqrt{112}}{\sqrt{185}}.

$$2+\frac{\sqrt{112}}{\sqrt{185}}$$

Factor 112=4^{2}\times 7. Rewrite the square root of the product \sqrt{4^{2}\times 7} as the product of square roots \sqrt{4^{2}}\sqrt{7}. Take the square root of 4^{2}.

$$2+\frac{4\sqrt{7}}{\sqrt{185}}$$

Rationalize the denominator of \frac{4\sqrt{7}}{\sqrt{185}} by multiplying numerator and denominator by \sqrt{185}.

$$2+\frac{4\sqrt{7}\sqrt{185}}{\left(\sqrt{185}\right)^{2}}$$

The square of \sqrt{185} is 185.

$$2+\frac{4\sqrt{7}\sqrt{185}}{185}$$

To multiply \sqrt{7} and \sqrt{185}, multiply the numbers under the square root.

$$2+\frac{4\sqrt{1295}}{185}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 2 times \frac{185}{185}.

$$\frac{2\times 185}{185}+\frac{4\sqrt{1295}}{185}$$

Since \frac{2\times 185}{185} and \frac{4\sqrt{1295}}{185} have the same denominator, add them by adding their numerators.

$$\frac{2\times 185+4\sqrt{1295}}{185}$$

Do the multiplications in 2\times 185+4\sqrt{1295}.

$$\frac{370+4\sqrt{1295}}{185}$$