Solve 20-sqrt{2/3div(17/27-1/3)+1/2div(frac{1}{6}-frac{1}{24})}-frac{3}{5}*(frac{7}{8}+frac{13}{8})

$$20 - \sqrt { \frac { 2 } { 3 } \div ( \frac { 17 } { 27 } - \frac { 1 } { 3 } ) + \frac { 1 } { 2 } \div ( \frac { 1 } { 6 } - \frac { 1 } { 24 } ) } - \frac { 3 } { 5 } * ( \frac { 7 } { 8 } + \frac { 13 } { 8 } )$$

Evaluate

$$16$$

Solution Steps

Least common multiple of 27 and 3 is 27. Convert \frac{17}{27} and \frac{1}{3} to fractions with denominator 27.

$$20-\sqrt{\frac{\frac{2}{3}}{\frac{17}{27}-\frac{9}{27}}+\frac{\frac{1}{2}}{\frac{1}{6}-\frac{1}{24}}}-\frac{3}{5}\left(\frac{7}{8}+\frac{13}{8}\right)$$

Since \frac{17}{27} and \frac{9}{27} have the same denominator, subtract them by subtracting their numerators.

$$20-\sqrt{\frac{\frac{2}{3}}{\frac{17-9}{27}}+\frac{\frac{1}{2}}{\frac{1}{6}-\frac{1}{24}}}-\frac{3}{5}\left(\frac{7}{8}+\frac{13}{8}\right)$$

Subtract 9 from 17 to get 8.

$$20-\sqrt{\frac{\frac{2}{3}}{\frac{8}{27}}+\frac{\frac{1}{2}}{\frac{1}{6}-\frac{1}{24}}}-\frac{3}{5}\left(\frac{7}{8}+\frac{13}{8}\right)$$

Divide \frac{2}{3} by \frac{8}{27} by multiplying \frac{2}{3} by the reciprocal of \frac{8}{27}.

$$20-\sqrt{\frac{2}{3}\times \frac{27}{8}+\frac{\frac{1}{2}}{\frac{1}{6}-\frac{1}{24}}}-\frac{3}{5}\left(\frac{7}{8}+\frac{13}{8}\right)$$

Multiply \frac{2}{3} times \frac{27}{8} by multiplying numerator times numerator and denominator times denominator.

$$20-\sqrt{\frac{2\times 27}{3\times 8}+\frac{\frac{1}{2}}{\frac{1}{6}-\frac{1}{24}}}-\frac{3}{5}\left(\frac{7}{8}+\frac{13}{8}\right)$$

Do the multiplications in the fraction \frac{2\times 27}{3\times 8}.

$$20-\sqrt{\frac{54}{24}+\frac{\frac{1}{2}}{\frac{1}{6}-\frac{1}{24}}}-\frac{3}{5}\left(\frac{7}{8}+\frac{13}{8}\right)$$

Reduce the fraction \frac{54}{24} to lowest terms by extracting and canceling out 6.

$$20-\sqrt{\frac{9}{4}+\frac{\frac{1}{2}}{\frac{1}{6}-\frac{1}{24}}}-\frac{3}{5}\left(\frac{7}{8}+\frac{13}{8}\right)$$

Least common multiple of 6 and 24 is 24. Convert \frac{1}{6} and \frac{1}{24} to fractions with denominator 24.

$$20-\sqrt{\frac{9}{4}+\frac{\frac{1}{2}}{\frac{4}{24}-\frac{1}{24}}}-\frac{3}{5}\left(\frac{7}{8}+\frac{13}{8}\right)$$

Since \frac{4}{24} and \frac{1}{24} have the same denominator, subtract them by subtracting their numerators.

$$20-\sqrt{\frac{9}{4}+\frac{\frac{1}{2}}{\frac{4-1}{24}}}-\frac{3}{5}\left(\frac{7}{8}+\frac{13}{8}\right)$$

Subtract 1 from 4 to get 3.

$$20-\sqrt{\frac{9}{4}+\frac{\frac{1}{2}}{\frac{3}{24}}}-\frac{3}{5}\left(\frac{7}{8}+\frac{13}{8}\right)$$

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

$$20-\sqrt{\frac{9}{4}+\frac{\frac{1}{2}}{\frac{1}{8}}}-\frac{3}{5}\left(\frac{7}{8}+\frac{13}{8}\right)$$

Divide \frac{1}{2} by \frac{1}{8} by multiplying \frac{1}{2} by the reciprocal of \frac{1}{8}.

$$20-\sqrt{\frac{9}{4}+\frac{1}{2}\times 8}-\frac{3}{5}\left(\frac{7}{8}+\frac{13}{8}\right)$$

Multiply \frac{1}{2} and 8 to get \frac{8}{2}.

$$20-\sqrt{\frac{9}{4}+\frac{8}{2}}-\frac{3}{5}\left(\frac{7}{8}+\frac{13}{8}\right)$$

Divide 8 by 2 to get 4.

$$20-\sqrt{\frac{9}{4}+4}-\frac{3}{5}\left(\frac{7}{8}+\frac{13}{8}\right)$$

Convert 4 to fraction \frac{16}{4}.

$$20-\sqrt{\frac{9}{4}+\frac{16}{4}}-\frac{3}{5}\left(\frac{7}{8}+\frac{13}{8}\right)$$

Since \frac{9}{4} and \frac{16}{4} have the same denominator, add them by adding their numerators.

$$20-\sqrt{\frac{9+16}{4}}-\frac{3}{5}\left(\frac{7}{8}+\frac{13}{8}\right)$$

Add 9 and 16 to get 25.

$$20-\sqrt{\frac{25}{4}}-\frac{3}{5}\left(\frac{7}{8}+\frac{13}{8}\right)$$

Rewrite the square root of the division \frac{25}{4} as the division of square roots \frac{\sqrt{25}}{\sqrt{4}}. Take the square root of both numerator and denominator.

$$20-\frac{5}{2}-\frac{3}{5}\left(\frac{7}{8}+\frac{13}{8}\right)$$

Convert 20 to fraction \frac{40}{2}.

$$\frac{40}{2}-\frac{5}{2}-\frac{3}{5}\left(\frac{7}{8}+\frac{13}{8}\right)$$

Since \frac{40}{2} and \frac{5}{2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{40-5}{2}-\frac{3}{5}\left(\frac{7}{8}+\frac{13}{8}\right)$$

Subtract 5 from 40 to get 35.

$$\frac{35}{2}-\frac{3}{5}\left(\frac{7}{8}+\frac{13}{8}\right)$$

Since \frac{7}{8} and \frac{13}{8} have the same denominator, add them by adding their numerators.

$$\frac{35}{2}-\frac{3}{5}\times \frac{7+13}{8}$$

Add 7 and 13 to get 20.

$$\frac{35}{2}-\frac{3}{5}\times \frac{20}{8}$$

Reduce the fraction \frac{20}{8} to lowest terms by extracting and canceling out 4.

$$\frac{35}{2}-\frac{3}{5}\times \frac{5}{2}$$

Multiply \frac{3}{5} times \frac{5}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{35}{2}-\frac{3\times 5}{5\times 2}$$

Cancel out 5 in both numerator and denominator.

$$\frac{35}{2}-\frac{3}{2}$$

Since \frac{35}{2} and \frac{3}{2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{35-3}{2}$$

Subtract 3 from 35 to get 32.

$$\frac{32}{2}$$

Divide 32 by 2 to get 16.

$$16$$

Factor

$$2^{4}$$