Solve sqrt{(3/2-1/6-4/5)+(3/8*frac{4}{9}+1)*frac{9}{2}*frac{16}{5}-frac{40}{3}}

$$\sqrt { ( \frac { 3 } { 2 } - \frac { 1 } { 6 } - \frac { 4 } { 5 } ) + ( \frac { 3 } { 8 } \times \frac { 4 } { 9 } + 1 ) \times \frac { 9 } { 2 } \times \frac { 16 } { 5 } - \frac { 40 } { 3 } }$$

Evaluate

$$2$$

Solution Steps

Least common multiple of 2 and 6 is 6. Convert \frac{3}{2} and \frac{1}{6} to fractions with denominator 6.

$$\sqrt{\frac{9}{6}-\frac{1}{6}-\frac{4}{5}+\left(\frac{3}{8}\times \frac{4}{9}+1\right)\times \frac{9}{2}\times \frac{16}{5}-\frac{40}{3}}$$

Since \frac{9}{6} and \frac{1}{6} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{9-1}{6}-\frac{4}{5}+\left(\frac{3}{8}\times \frac{4}{9}+1\right)\times \frac{9}{2}\times \frac{16}{5}-\frac{40}{3}}$$

Subtract 1 from 9 to get 8.

$$\sqrt{\frac{8}{6}-\frac{4}{5}+\left(\frac{3}{8}\times \frac{4}{9}+1\right)\times \frac{9}{2}\times \frac{16}{5}-\frac{40}{3}}$$

Reduce the fraction \frac{8}{6} to lowest terms by extracting and canceling out 2.

$$\sqrt{\frac{4}{3}-\frac{4}{5}+\left(\frac{3}{8}\times \frac{4}{9}+1\right)\times \frac{9}{2}\times \frac{16}{5}-\frac{40}{3}}$$

Least common multiple of 3 and 5 is 15. Convert \frac{4}{3} and \frac{4}{5} to fractions with denominator 15.

$$\sqrt{\frac{20}{15}-\frac{12}{15}+\left(\frac{3}{8}\times \frac{4}{9}+1\right)\times \frac{9}{2}\times \frac{16}{5}-\frac{40}{3}}$$

Since \frac{20}{15} and \frac{12}{15} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{20-12}{15}+\left(\frac{3}{8}\times \frac{4}{9}+1\right)\times \frac{9}{2}\times \frac{16}{5}-\frac{40}{3}}$$

Subtract 12 from 20 to get 8.

$$\sqrt{\frac{8}{15}+\left(\frac{3}{8}\times \frac{4}{9}+1\right)\times \frac{9}{2}\times \frac{16}{5}-\frac{40}{3}}$$

Multiply \frac{3}{8} times \frac{4}{9} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{8}{15}+\left(\frac{3\times 4}{8\times 9}+1\right)\times \frac{9}{2}\times \frac{16}{5}-\frac{40}{3}}$$

Do the multiplications in the fraction \frac{3\times 4}{8\times 9}.

$$\sqrt{\frac{8}{15}+\left(\frac{12}{72}+1\right)\times \frac{9}{2}\times \frac{16}{5}-\frac{40}{3}}$$

Reduce the fraction \frac{12}{72} to lowest terms by extracting and canceling out 12.

$$\sqrt{\frac{8}{15}+\left(\frac{1}{6}+1\right)\times \frac{9}{2}\times \frac{16}{5}-\frac{40}{3}}$$

Convert 1 to fraction \frac{6}{6}.

$$\sqrt{\frac{8}{15}+\left(\frac{1}{6}+\frac{6}{6}\right)\times \frac{9}{2}\times \frac{16}{5}-\frac{40}{3}}$$

Since \frac{1}{6} and \frac{6}{6} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{8}{15}+\frac{1+6}{6}\times \frac{9}{2}\times \frac{16}{5}-\frac{40}{3}}$$

Add 1 and 6 to get 7.

$$\sqrt{\frac{8}{15}+\frac{7}{6}\times \frac{9}{2}\times \frac{16}{5}-\frac{40}{3}}$$

Multiply \frac{7}{6} times \frac{9}{2} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{8}{15}+\frac{7\times 9}{6\times 2}\times \frac{16}{5}-\frac{40}{3}}$$

Do the multiplications in the fraction \frac{7\times 9}{6\times 2}.

$$\sqrt{\frac{8}{15}+\frac{63}{12}\times \frac{16}{5}-\frac{40}{3}}$$

Reduce the fraction \frac{63}{12} to lowest terms by extracting and canceling out 3.

$$\sqrt{\frac{8}{15}+\frac{21}{4}\times \frac{16}{5}-\frac{40}{3}}$$

Multiply \frac{21}{4} times \frac{16}{5} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{8}{15}+\frac{21\times 16}{4\times 5}-\frac{40}{3}}$$

Do the multiplications in the fraction \frac{21\times 16}{4\times 5}.

$$\sqrt{\frac{8}{15}+\frac{336}{20}-\frac{40}{3}}$$

Reduce the fraction \frac{336}{20} to lowest terms by extracting and canceling out 4.

$$\sqrt{\frac{8}{15}+\frac{84}{5}-\frac{40}{3}}$$

Least common multiple of 15 and 5 is 15. Convert \frac{8}{15} and \frac{84}{5} to fractions with denominator 15.

$$\sqrt{\frac{8}{15}+\frac{252}{15}-\frac{40}{3}}$$

Since \frac{8}{15} and \frac{252}{15} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{8+252}{15}-\frac{40}{3}}$$

Add 8 and 252 to get 260.

$$\sqrt{\frac{260}{15}-\frac{40}{3}}$$

Reduce the fraction \frac{260}{15} to lowest terms by extracting and canceling out 5.

$$\sqrt{\frac{52}{3}-\frac{40}{3}}$$

Since \frac{52}{3} and \frac{40}{3} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{52-40}{3}}$$

Subtract 40 from 52 to get 12.

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

Divide 12 by 3 to get 4.

$$\sqrt{4}$$

Calculate the square root of 4 and get 2.

$$2$$

Factor

$$2$$