Solve sqrt{{2/3+5/4:[1+3/4*(1/3+frac{1}{4}*frac{12}{7})-frac{1}{7}]}*frac{3}{37}+frac{1}{64}}

$$\sqrt { \{ \frac { 2 } { 3 } + \frac { 5 } { 4 } : [ 1 + \frac { 3 } { 4 } \times ( \frac { 1 } { 3 } + \frac { 1 } { 4 } \times \frac { 12 } { 7 } ) - \frac { 1 } { 7 } ] \} \times \frac { 3 } { 37 } + \frac { 1 } { 64 } }$$

Evaluate

$$\frac{3}{8}=0.375$$

Solution Steps

Multiply \frac{1}{4} times \frac{12}{7} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\left(\frac{2}{3}+\frac{\frac{5}{4}}{1+\frac{3}{4}\left(\frac{1}{3}+\frac{1\times 12}{4\times 7}\right)-\frac{1}{7}}\right)\times \frac{3}{37}+\frac{1}{64}}$$

Do the multiplications in the fraction \frac{1\times 12}{4\times 7}.

$$\sqrt{\left(\frac{2}{3}+\frac{\frac{5}{4}}{1+\frac{3}{4}\left(\frac{1}{3}+\frac{12}{28}\right)-\frac{1}{7}}\right)\times \frac{3}{37}+\frac{1}{64}}$$

Reduce the fraction \frac{12}{28} to lowest terms by extracting and canceling out 4.

$$\sqrt{\left(\frac{2}{3}+\frac{\frac{5}{4}}{1+\frac{3}{4}\left(\frac{1}{3}+\frac{3}{7}\right)-\frac{1}{7}}\right)\times \frac{3}{37}+\frac{1}{64}}$$

Least common multiple of 3 and 7 is 21. Convert \frac{1}{3} and \frac{3}{7} to fractions with denominator 21.

$$\sqrt{\left(\frac{2}{3}+\frac{\frac{5}{4}}{1+\frac{3}{4}\left(\frac{7}{21}+\frac{9}{21}\right)-\frac{1}{7}}\right)\times \frac{3}{37}+\frac{1}{64}}$$

Since \frac{7}{21} and \frac{9}{21} have the same denominator, add them by adding their numerators.

$$\sqrt{\left(\frac{2}{3}+\frac{\frac{5}{4}}{1+\frac{3}{4}\times \frac{7+9}{21}-\frac{1}{7}}\right)\times \frac{3}{37}+\frac{1}{64}}$$

Add 7 and 9 to get 16.

$$\sqrt{\left(\frac{2}{3}+\frac{\frac{5}{4}}{1+\frac{3}{4}\times \frac{16}{21}-\frac{1}{7}}\right)\times \frac{3}{37}+\frac{1}{64}}$$

Multiply \frac{3}{4} times \frac{16}{21} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\left(\frac{2}{3}+\frac{\frac{5}{4}}{1+\frac{3\times 16}{4\times 21}-\frac{1}{7}}\right)\times \frac{3}{37}+\frac{1}{64}}$$

Do the multiplications in the fraction \frac{3\times 16}{4\times 21}.

$$\sqrt{\left(\frac{2}{3}+\frac{\frac{5}{4}}{1+\frac{48}{84}-\frac{1}{7}}\right)\times \frac{3}{37}+\frac{1}{64}}$$

Reduce the fraction \frac{48}{84} to lowest terms by extracting and canceling out 12.

$$\sqrt{\left(\frac{2}{3}+\frac{\frac{5}{4}}{1+\frac{4}{7}-\frac{1}{7}}\right)\times \frac{3}{37}+\frac{1}{64}}$$

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

$$\sqrt{\left(\frac{2}{3}+\frac{\frac{5}{4}}{\frac{7}{7}+\frac{4}{7}-\frac{1}{7}}\right)\times \frac{3}{37}+\frac{1}{64}}$$

Since \frac{7}{7} and \frac{4}{7} have the same denominator, add them by adding their numerators.

$$\sqrt{\left(\frac{2}{3}+\frac{\frac{5}{4}}{\frac{7+4}{7}-\frac{1}{7}}\right)\times \frac{3}{37}+\frac{1}{64}}$$

Add 7 and 4 to get 11.

$$\sqrt{\left(\frac{2}{3}+\frac{\frac{5}{4}}{\frac{11}{7}-\frac{1}{7}}\right)\times \frac{3}{37}+\frac{1}{64}}$$

Since \frac{11}{7} and \frac{1}{7} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\left(\frac{2}{3}+\frac{\frac{5}{4}}{\frac{11-1}{7}}\right)\times \frac{3}{37}+\frac{1}{64}}$$

Subtract 1 from 11 to get 10.

$$\sqrt{\left(\frac{2}{3}+\frac{\frac{5}{4}}{\frac{10}{7}}\right)\times \frac{3}{37}+\frac{1}{64}}$$

Divide \frac{5}{4} by \frac{10}{7} by multiplying \frac{5}{4} by the reciprocal of \frac{10}{7}.

$$\sqrt{\left(\frac{2}{3}+\frac{5}{4}\times \frac{7}{10}\right)\times \frac{3}{37}+\frac{1}{64}}$$

Multiply \frac{5}{4} times \frac{7}{10} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\left(\frac{2}{3}+\frac{5\times 7}{4\times 10}\right)\times \frac{3}{37}+\frac{1}{64}}$$

Do the multiplications in the fraction \frac{5\times 7}{4\times 10}.

$$\sqrt{\left(\frac{2}{3}+\frac{35}{40}\right)\times \frac{3}{37}+\frac{1}{64}}$$

Reduce the fraction \frac{35}{40} to lowest terms by extracting and canceling out 5.

$$\sqrt{\left(\frac{2}{3}+\frac{7}{8}\right)\times \frac{3}{37}+\frac{1}{64}}$$

Least common multiple of 3 and 8 is 24. Convert \frac{2}{3} and \frac{7}{8} to fractions with denominator 24.

$$\sqrt{\left(\frac{16}{24}+\frac{21}{24}\right)\times \frac{3}{37}+\frac{1}{64}}$$

Since \frac{16}{24} and \frac{21}{24} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{16+21}{24}\times \frac{3}{37}+\frac{1}{64}}$$

Add 16 and 21 to get 37.

$$\sqrt{\frac{37}{24}\times \frac{3}{37}+\frac{1}{64}}$$

Multiply \frac{37}{24} times \frac{3}{37} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{37\times 3}{24\times 37}+\frac{1}{64}}$$

Cancel out 37 in both numerator and denominator.

$$\sqrt{\frac{3}{24}+\frac{1}{64}}$$

Reduce the fraction \frac{3}{24} to lowest terms by extracting and canceling out 3.

$$\sqrt{\frac{1}{8}+\frac{1}{64}}$$

Least common multiple of 8 and 64 is 64. Convert \frac{1}{8} and \frac{1}{64} to fractions with denominator 64.

$$\sqrt{\frac{8}{64}+\frac{1}{64}}$$

Since \frac{8}{64} and \frac{1}{64} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{8+1}{64}}$$

Add 8 and 1 to get 9.

$$\sqrt{\frac{9}{64}}$$

Rewrite the square root of the division \frac{9}{64} as the division of square roots \frac{\sqrt{9}}{\sqrt{64}}. Take the square root of both numerator and denominator.

$$\frac{3}{8}$$

Factor

$$\frac{3}{2 ^ {3}} = 0.375$$