Solve 2-[sqrt{16/9}-(1/2+2/5)-1/3]-(frac{4}{3}+2)-frac{1}{5}=

$$2 - [ \sqrt { \frac { 16 } { 9 } } - ( \frac { 1 } { 2 } + \frac { 2 } { 5 } ) - \frac { 1 } { 3 } ] - ( \frac { 4 } { 3 } + 2 ) - \frac { 1 } { 5 } =$$

Evaluate

$$-\frac{49}{30}\approx -1.633333333$$

Solution Steps

Rewrite the square root of the division \frac{16}{9} as the division of square roots \frac{\sqrt{16}}{\sqrt{9}}. Take the square root of both numerator and denominator.

$$2-\left(\frac{4}{3}-\left(\frac{1}{2}+\frac{2}{5}\right)-\frac{1}{3}\right)-\left(\frac{4}{3}+2\right)-\frac{1}{5}$$

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

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

Since \frac{5}{10} and \frac{4}{10} have the same denominator, add them by adding their numerators.

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

Add 5 and 4 to get 9.

$$2-\left(\frac{4}{3}-\frac{9}{10}-\frac{1}{3}\right)-\left(\frac{4}{3}+2\right)-\frac{1}{5}$$

Least common multiple of 3 and 10 is 30. Convert \frac{4}{3} and \frac{9}{10} to fractions with denominator 30.

$$2-\left(\frac{40}{30}-\frac{27}{30}-\frac{1}{3}\right)-\left(\frac{4}{3}+2\right)-\frac{1}{5}$$

Since \frac{40}{30} and \frac{27}{30} have the same denominator, subtract them by subtracting their numerators.

$$2-\left(\frac{40-27}{30}-\frac{1}{3}\right)-\left(\frac{4}{3}+2\right)-\frac{1}{5}$$

Subtract 27 from 40 to get 13.

$$2-\left(\frac{13}{30}-\frac{1}{3}\right)-\left(\frac{4}{3}+2\right)-\frac{1}{5}$$

Least common multiple of 30 and 3 is 30. Convert \frac{13}{30} and \frac{1}{3} to fractions with denominator 30.

$$2-\left(\frac{13}{30}-\frac{10}{30}\right)-\left(\frac{4}{3}+2\right)-\frac{1}{5}$$

Since \frac{13}{30} and \frac{10}{30} have the same denominator, subtract them by subtracting their numerators.

$$2-\frac{13-10}{30}-\left(\frac{4}{3}+2\right)-\frac{1}{5}$$

Subtract 10 from 13 to get 3.

$$2-\frac{3}{30}-\left(\frac{4}{3}+2\right)-\frac{1}{5}$$

Reduce the fraction \frac{3}{30} to lowest terms by extracting and canceling out 3.

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

Convert 2 to fraction \frac{20}{10}.

$$\frac{20}{10}-\frac{1}{10}-\left(\frac{4}{3}+2\right)-\frac{1}{5}$$

Since \frac{20}{10} and \frac{1}{10} have the same denominator, subtract them by subtracting their numerators.

$$\frac{20-1}{10}-\left(\frac{4}{3}+2\right)-\frac{1}{5}$$

Subtract 1 from 20 to get 19.

$$\frac{19}{10}-\left(\frac{4}{3}+2\right)-\frac{1}{5}$$

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

$$\frac{19}{10}-\left(\frac{4}{3}+\frac{6}{3}\right)-\frac{1}{5}$$

Since \frac{4}{3} and \frac{6}{3} have the same denominator, add them by adding their numerators.

$$\frac{19}{10}-\frac{4+6}{3}-\frac{1}{5}$$

Add 4 and 6 to get 10.

$$\frac{19}{10}-\frac{10}{3}-\frac{1}{5}$$

Least common multiple of 10 and 3 is 30. Convert \frac{19}{10} and \frac{10}{3} to fractions with denominator 30.

$$\frac{57}{30}-\frac{100}{30}-\frac{1}{5}$$

Since \frac{57}{30} and \frac{100}{30} have the same denominator, subtract them by subtracting their numerators.

$$\frac{57-100}{30}-\frac{1}{5}$$

Subtract 100 from 57 to get -43.

$$-\frac{43}{30}-\frac{1}{5}$$

Least common multiple of 30 and 5 is 30. Convert -\frac{43}{30} and \frac{1}{5} to fractions with denominator 30.

$$-\frac{43}{30}-\frac{6}{30}$$

Since -\frac{43}{30} and \frac{6}{30} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-43-6}{30}$$

Subtract 6 from -43 to get -49.

$$-\frac{49}{30}$$

Factor

$$-\frac{49}{30} = -1\frac{19}{30} = -1.6333333333333333$$