Solve sqrt{(5/3-3/5):[7/9-13/15:(frac{4}{5}+frac{1}{2})+frac{1}{3}]*frac{5}{3}}=

$$\sqrt { ( \frac { 5 } { 3 } - \frac { 3 } { 5 } ) : [ \frac { 7 } { 9 } - \frac { 13 } { 15 } : ( \frac { 4 } { 5 } + \frac { 1 } { 2 } ) + \frac { 1 } { 3 } ] \times \frac { 5 } { 3 } } =$$

Evaluate

$$2$$

Solution Steps

Least common multiple of 3 and 5 is 15. Convert \frac{5}{3} and \frac{3}{5} to fractions with denominator 15.

$$\sqrt{\frac{\frac{25}{15}-\frac{9}{15}}{\frac{7}{9}-\frac{\frac{13}{15}}{\frac{4}{5}+\frac{1}{2}}+\frac{1}{3}}\times \frac{5}{3}}$$

Since \frac{25}{15} and \frac{9}{15} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{\frac{25-9}{15}}{\frac{7}{9}-\frac{\frac{13}{15}}{\frac{4}{5}+\frac{1}{2}}+\frac{1}{3}}\times \frac{5}{3}}$$

Subtract 9 from 25 to get 16.

$$\sqrt{\frac{\frac{16}{15}}{\frac{7}{9}-\frac{\frac{13}{15}}{\frac{4}{5}+\frac{1}{2}}+\frac{1}{3}}\times \frac{5}{3}}$$

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

$$\sqrt{\frac{\frac{16}{15}}{\frac{7}{9}-\frac{\frac{13}{15}}{\frac{8}{10}+\frac{5}{10}}+\frac{1}{3}}\times \frac{5}{3}}$$

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

$$\sqrt{\frac{\frac{16}{15}}{\frac{7}{9}-\frac{\frac{13}{15}}{\frac{8+5}{10}}+\frac{1}{3}}\times \frac{5}{3}}$$

Add 8 and 5 to get 13.

$$\sqrt{\frac{\frac{16}{15}}{\frac{7}{9}-\frac{\frac{13}{15}}{\frac{13}{10}}+\frac{1}{3}}\times \frac{5}{3}}$$

Divide \frac{13}{15} by \frac{13}{10} by multiplying \frac{13}{15} by the reciprocal of \frac{13}{10}.

$$\sqrt{\frac{\frac{16}{15}}{\frac{7}{9}-\frac{13}{15}\times \frac{10}{13}+\frac{1}{3}}\times \frac{5}{3}}$$

Multiply \frac{13}{15} times \frac{10}{13} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{\frac{16}{15}}{\frac{7}{9}-\frac{13\times 10}{15\times 13}+\frac{1}{3}}\times \frac{5}{3}}$$

Cancel out 13 in both numerator and denominator.

$$\sqrt{\frac{\frac{16}{15}}{\frac{7}{9}-\frac{10}{15}+\frac{1}{3}}\times \frac{5}{3}}$$

Reduce the fraction \frac{10}{15} to lowest terms by extracting and canceling out 5.

$$\sqrt{\frac{\frac{16}{15}}{\frac{7}{9}-\frac{2}{3}+\frac{1}{3}}\times \frac{5}{3}}$$

Least common multiple of 9 and 3 is 9. Convert \frac{7}{9} and \frac{2}{3} to fractions with denominator 9.

$$\sqrt{\frac{\frac{16}{15}}{\frac{7}{9}-\frac{6}{9}+\frac{1}{3}}\times \frac{5}{3}}$$

Since \frac{7}{9} and \frac{6}{9} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{\frac{16}{15}}{\frac{7-6}{9}+\frac{1}{3}}\times \frac{5}{3}}$$

Subtract 6 from 7 to get 1.

$$\sqrt{\frac{\frac{16}{15}}{\frac{1}{9}+\frac{1}{3}}\times \frac{5}{3}}$$

Least common multiple of 9 and 3 is 9. Convert \frac{1}{9} and \frac{1}{3} to fractions with denominator 9.

$$\sqrt{\frac{\frac{16}{15}}{\frac{1}{9}+\frac{3}{9}}\times \frac{5}{3}}$$

Since \frac{1}{9} and \frac{3}{9} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{\frac{16}{15}}{\frac{1+3}{9}}\times \frac{5}{3}}$$

Add 1 and 3 to get 4.

$$\sqrt{\frac{\frac{16}{15}}{\frac{4}{9}}\times \frac{5}{3}}$$

Divide \frac{16}{15} by \frac{4}{9} by multiplying \frac{16}{15} by the reciprocal of \frac{4}{9}.

$$\sqrt{\frac{16}{15}\times \frac{9}{4}\times \frac{5}{3}}$$

Multiply \frac{16}{15} times \frac{9}{4} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{16\times 9}{15\times 4}\times \frac{5}{3}}$$

Do the multiplications in the fraction \frac{16\times 9}{15\times 4}.

$$\sqrt{\frac{144}{60}\times \frac{5}{3}}$$

Reduce the fraction \frac{144}{60} to lowest terms by extracting and canceling out 12.

$$\sqrt{\frac{12}{5}\times \frac{5}{3}}$$

Multiply \frac{12}{5} times \frac{5}{3} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{12\times 5}{5\times 3}}$$

Cancel out 5 in both numerator and denominator.

$$\sqrt{\frac{12}{3}}$$

Divide 12 by 3 to get 4.

$$\sqrt{4}$$

Calculate the square root of 4 and get 2.

$$2$$

Factor

$$2$$