Solve sqrt{11/3-5/6-2/3/1-frac{5{7}}left(1-frac{33}{49}right)}+sqrt{frac{frac{frac{1}{2}+frac{5}{8}}{frac{5}{4}+1}}{2}}

$$\sqrt{ \frac{ \frac{ 11 }{ 3 } - \frac{ 5 }{ 6 } - \frac{ 2 }{ 3 } }{ 1- \frac{ 5 }{ 7 } } \left( 1- \frac{ 33 }{ 49 } \right) } + \sqrt{ \frac{ \frac{ \frac{ 1 }{ 2 } + \frac{ 5 }{ 8 } }{ \frac{ 5 }{ 4 } +1 } }{ 2 } }$$

Evaluate

$$\frac{2\sqrt{273}}{21}+\frac{1}{2}\approx 2.073591585$$

Short Solution Steps

Least common multiple of 3 and 6 is 6. Convert \frac{11}{3} and \frac{5}{6} to fractions with denominator 6.

$$\sqrt{\frac{\frac{22}{6}-\frac{5}{6}-\frac{2}{3}}{1-\frac{5}{7}}\left(1-\frac{33}{49}\right)}+\sqrt{\frac{\frac{\frac{1}{2}+\frac{5}{8}}{\frac{5}{4}+1}}{2}}$$

Since \frac{22}{6} and \frac{5}{6} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{\frac{22-5}{6}-\frac{2}{3}}{1-\frac{5}{7}}\left(1-\frac{33}{49}\right)}+\sqrt{\frac{\frac{\frac{1}{2}+\frac{5}{8}}{\frac{5}{4}+1}}{2}}$$

Subtract 5 from 22 to get 17.

$$\sqrt{\frac{\frac{17}{6}-\frac{2}{3}}{1-\frac{5}{7}}\left(1-\frac{33}{49}\right)}+\sqrt{\frac{\frac{\frac{1}{2}+\frac{5}{8}}{\frac{5}{4}+1}}{2}}$$

Least common multiple of 6 and 3 is 6. Convert \frac{17}{6} and \frac{2}{3} to fractions with denominator 6.

$$\sqrt{\frac{\frac{17}{6}-\frac{4}{6}}{1-\frac{5}{7}}\left(1-\frac{33}{49}\right)}+\sqrt{\frac{\frac{\frac{1}{2}+\frac{5}{8}}{\frac{5}{4}+1}}{2}}$$

Since \frac{17}{6} and \frac{4}{6} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{\frac{17-4}{6}}{1-\frac{5}{7}}\left(1-\frac{33}{49}\right)}+\sqrt{\frac{\frac{\frac{1}{2}+\frac{5}{8}}{\frac{5}{4}+1}}{2}}$$

Subtract 4 from 17 to get 13.

$$\sqrt{\frac{\frac{13}{6}}{1-\frac{5}{7}}\left(1-\frac{33}{49}\right)}+\sqrt{\frac{\frac{\frac{1}{2}+\frac{5}{8}}{\frac{5}{4}+1}}{2}}$$

Convert 1 to fraction \frac{7}{7}.

$$\sqrt{\frac{\frac{13}{6}}{\frac{7}{7}-\frac{5}{7}}\left(1-\frac{33}{49}\right)}+\sqrt{\frac{\frac{\frac{1}{2}+\frac{5}{8}}{\frac{5}{4}+1}}{2}}$$

Since \frac{7}{7} and \frac{5}{7} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{\frac{13}{6}}{\frac{7-5}{7}}\left(1-\frac{33}{49}\right)}+\sqrt{\frac{\frac{\frac{1}{2}+\frac{5}{8}}{\frac{5}{4}+1}}{2}}$$

Subtract 5 from 7 to get 2.

$$\sqrt{\frac{\frac{13}{6}}{\frac{2}{7}}\left(1-\frac{33}{49}\right)}+\sqrt{\frac{\frac{\frac{1}{2}+\frac{5}{8}}{\frac{5}{4}+1}}{2}}$$

Divide \frac{13}{6} by \frac{2}{7} by multiplying \frac{13}{6} by the reciprocal of \frac{2}{7}.

$$\sqrt{\frac{13}{6}\times \frac{7}{2}\left(1-\frac{33}{49}\right)}+\sqrt{\frac{\frac{\frac{1}{2}+\frac{5}{8}}{\frac{5}{4}+1}}{2}}$$

Multiply \frac{13}{6} times \frac{7}{2} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{13\times 7}{6\times 2}\left(1-\frac{33}{49}\right)}+\sqrt{\frac{\frac{\frac{1}{2}+\frac{5}{8}}{\frac{5}{4}+1}}{2}}$$

Do the multiplications in the fraction \frac{13\times 7}{6\times 2}.

$$\sqrt{\frac{91}{12}\left(1-\frac{33}{49}\right)}+\sqrt{\frac{\frac{\frac{1}{2}+\frac{5}{8}}{\frac{5}{4}+1}}{2}}$$

Convert 1 to fraction \frac{49}{49}.

$$\sqrt{\frac{91}{12}\left(\frac{49}{49}-\frac{33}{49}\right)}+\sqrt{\frac{\frac{\frac{1}{2}+\frac{5}{8}}{\frac{5}{4}+1}}{2}}$$

Since \frac{49}{49} and \frac{33}{49} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{91}{12}\times \frac{49-33}{49}}+\sqrt{\frac{\frac{\frac{1}{2}+\frac{5}{8}}{\frac{5}{4}+1}}{2}}$$

Subtract 33 from 49 to get 16.

$$\sqrt{\frac{91}{12}\times \frac{16}{49}}+\sqrt{\frac{\frac{\frac{1}{2}+\frac{5}{8}}{\frac{5}{4}+1}}{2}}$$

Multiply \frac{91}{12} times \frac{16}{49} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{91\times 16}{12\times 49}}+\sqrt{\frac{\frac{\frac{1}{2}+\frac{5}{8}}{\frac{5}{4}+1}}{2}}$$

Do the multiplications in the fraction \frac{91\times 16}{12\times 49}.

$$\sqrt{\frac{1456}{588}}+\sqrt{\frac{\frac{\frac{1}{2}+\frac{5}{8}}{\frac{5}{4}+1}}{2}}$$

Reduce the fraction \frac{1456}{588} to lowest terms by extracting and canceling out 28.

$$\sqrt{\frac{52}{21}}+\sqrt{\frac{\frac{\frac{1}{2}+\frac{5}{8}}{\frac{5}{4}+1}}{2}}$$

Rewrite the square root of the division \sqrt{\frac{52}{21}} as the division of square roots \frac{\sqrt{52}}{\sqrt{21}}.

$$\frac{\sqrt{52}}{\sqrt{21}}+\sqrt{\frac{\frac{\frac{1}{2}+\frac{5}{8}}{\frac{5}{4}+1}}{2}}$$

Factor 52=2^{2}\times 13. Rewrite the square root of the product \sqrt{2^{2}\times 13} as the product of square roots \sqrt{2^{2}}\sqrt{13}. Take the square root of 2^{2}.

$$\frac{2\sqrt{13}}{\sqrt{21}}+\sqrt{\frac{\frac{\frac{1}{2}+\frac{5}{8}}{\frac{5}{4}+1}}{2}}$$

Rationalize the denominator of \frac{2\sqrt{13}}{\sqrt{21}} by multiplying numerator and denominator by \sqrt{21}.

$$\frac{2\sqrt{13}\sqrt{21}}{\left(\sqrt{21}\right)^{2}}+\sqrt{\frac{\frac{\frac{1}{2}+\frac{5}{8}}{\frac{5}{4}+1}}{2}}$$

The square of \sqrt{21} is 21.

$$\frac{2\sqrt{13}\sqrt{21}}{21}+\sqrt{\frac{\frac{\frac{1}{2}+\frac{5}{8}}{\frac{5}{4}+1}}{2}}$$

To multiply \sqrt{13} and \sqrt{21}, multiply the numbers under the square root.

$$\frac{2\sqrt{273}}{21}+\sqrt{\frac{\frac{\frac{1}{2}+\frac{5}{8}}{\frac{5}{4}+1}}{2}}$$

Express \frac{\frac{\frac{1}{2}+\frac{5}{8}}{\frac{5}{4}+1}}{2} as a single fraction.

$$\frac{2\sqrt{273}}{21}+\sqrt{\frac{\frac{1}{2}+\frac{5}{8}}{\left(\frac{5}{4}+1\right)\times 2}}$$

Least common multiple of 2 and 8 is 8. Convert \frac{1}{2} and \frac{5}{8} to fractions with denominator 8.

$$\frac{2\sqrt{273}}{21}+\sqrt{\frac{\frac{4}{8}+\frac{5}{8}}{\left(\frac{5}{4}+1\right)\times 2}}$$

Since \frac{4}{8} and \frac{5}{8} have the same denominator, add them by adding their numerators.

$$\frac{2\sqrt{273}}{21}+\sqrt{\frac{\frac{4+5}{8}}{\left(\frac{5}{4}+1\right)\times 2}}$$

Add 4 and 5 to get 9.

$$\frac{2\sqrt{273}}{21}+\sqrt{\frac{\frac{9}{8}}{\left(\frac{5}{4}+1\right)\times 2}}$$

Convert 1 to fraction \frac{4}{4}.

$$\frac{2\sqrt{273}}{21}+\sqrt{\frac{\frac{9}{8}}{\left(\frac{5}{4}+\frac{4}{4}\right)\times 2}}$$

Since \frac{5}{4} and \frac{4}{4} have the same denominator, add them by adding their numerators.

$$\frac{2\sqrt{273}}{21}+\sqrt{\frac{\frac{9}{8}}{\frac{5+4}{4}\times 2}}$$

Add 5 and 4 to get 9.

$$\frac{2\sqrt{273}}{21}+\sqrt{\frac{\frac{9}{8}}{\frac{9}{4}\times 2}}$$

Express \frac{9}{4}\times 2 as a single fraction.

$$\frac{2\sqrt{273}}{21}+\sqrt{\frac{\frac{9}{8}}{\frac{9\times 2}{4}}}$$

Multiply 9 and 2 to get 18.

$$\frac{2\sqrt{273}}{21}+\sqrt{\frac{\frac{9}{8}}{\frac{18}{4}}}$$

Reduce the fraction \frac{18}{4} to lowest terms by extracting and canceling out 2.

$$\frac{2\sqrt{273}}{21}+\sqrt{\frac{\frac{9}{8}}{\frac{9}{2}}}$$

Divide \frac{9}{8} by \frac{9}{2} by multiplying \frac{9}{8} by the reciprocal of \frac{9}{2}.

$$\frac{2\sqrt{273}}{21}+\sqrt{\frac{9}{8}\times \frac{2}{9}}$$

Multiply \frac{9}{8} times \frac{2}{9} by multiplying numerator times numerator and denominator times denominator.

$$\frac{2\sqrt{273}}{21}+\sqrt{\frac{9\times 2}{8\times 9}}$$

Cancel out 9 in both numerator and denominator.

$$\frac{2\sqrt{273}}{21}+\sqrt{\frac{2}{8}}$$

Reduce the fraction \frac{2}{8} to lowest terms by extracting and canceling out 2.

$$\frac{2\sqrt{273}}{21}+\sqrt{\frac{1}{4}}$$

Rewrite the square root of the division \frac{1}{4} as the division of square roots \frac{\sqrt{1}}{\sqrt{4}}. Take the square root of both numerator and denominator.

$$\frac{2\sqrt{273}}{21}+\frac{1}{2}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 21 and 2 is 42. Multiply \frac{2\sqrt{273}}{21} times \frac{2}{2}. Multiply \frac{1}{2} times \frac{21}{21}.

$$\frac{2\times 2\sqrt{273}}{42}+\frac{21}{42}$$

Since \frac{2\times 2\sqrt{273}}{42} and \frac{21}{42} have the same denominator, add them by adding their numerators.

$$\frac{2\times 2\sqrt{273}+21}{42}$$

Do the multiplications in 2\times 2\sqrt{273}+21.

$$\frac{4\sqrt{273}+21}{42}$$