Solve sqrt{{[(10/3-11/6)*4/15+3/5*(frac{2}{3}-frac{1}{2})]:frac{8}{3}+1-(frac{1}{2})^2}*(3+frac{3}{4})}

$$\sqrt { \{ [ ( \frac { 10 } { 3 } - \frac { 11 } { 6 } ) \times \frac { 4 } { 15 } + \frac { 3 } { 5 } \times ( \frac { 2 } { 3 } - \frac { 1 } { 2 } ) ] : \frac { 8 } { 3 } + 1 - ( \frac { 1 } { 2 } ) ^ { 2 } \} \times ( 3 + \frac { 3 } { 4 } ) }$$

Evaluate

$$\frac{15}{8}=1.875$$

Solution Steps

Least common multiple of 3 and 6 is 6. Convert \frac{10}{3} and \frac{11}{6} to fractions with denominator 6.

$$\sqrt{\left(\frac{\left(\frac{20}{6}-\frac{11}{6}\right)\times \frac{4}{15}+\frac{3}{5}\left(\frac{2}{3}-\frac{1}{2}\right)}{\frac{8}{3}}+1-\left(\frac{1}{2}\right)^{2}\right)\left(3+\frac{3}{4}\right)}$$

Since \frac{20}{6} and \frac{11}{6} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\left(\frac{\frac{20-11}{6}\times \frac{4}{15}+\frac{3}{5}\left(\frac{2}{3}-\frac{1}{2}\right)}{\frac{8}{3}}+1-\left(\frac{1}{2}\right)^{2}\right)\left(3+\frac{3}{4}\right)}$$

Subtract 11 from 20 to get 9.

$$\sqrt{\left(\frac{\frac{9}{6}\times \frac{4}{15}+\frac{3}{5}\left(\frac{2}{3}-\frac{1}{2}\right)}{\frac{8}{3}}+1-\left(\frac{1}{2}\right)^{2}\right)\left(3+\frac{3}{4}\right)}$$

Reduce the fraction \frac{9}{6} to lowest terms by extracting and canceling out 3.

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

Multiply \frac{3}{2} times \frac{4}{15} by multiplying numerator times numerator and denominator times denominator.

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

Do the multiplications in the fraction \frac{3\times 4}{2\times 15}.

$$\sqrt{\left(\frac{\frac{12}{30}+\frac{3}{5}\left(\frac{2}{3}-\frac{1}{2}\right)}{\frac{8}{3}}+1-\left(\frac{1}{2}\right)^{2}\right)\left(3+\frac{3}{4}\right)}$$

Reduce the fraction \frac{12}{30} to lowest terms by extracting and canceling out 6.

$$\sqrt{\left(\frac{\frac{2}{5}+\frac{3}{5}\left(\frac{2}{3}-\frac{1}{2}\right)}{\frac{8}{3}}+1-\left(\frac{1}{2}\right)^{2}\right)\left(3+\frac{3}{4}\right)}$$

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

$$\sqrt{\left(\frac{\frac{2}{5}+\frac{3}{5}\left(\frac{4}{6}-\frac{3}{6}\right)}{\frac{8}{3}}+1-\left(\frac{1}{2}\right)^{2}\right)\left(3+\frac{3}{4}\right)}$$

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

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

Subtract 3 from 4 to get 1.

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

Multiply \frac{3}{5} times \frac{1}{6} by multiplying numerator times numerator and denominator times denominator.

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

Do the multiplications in the fraction \frac{3\times 1}{5\times 6}.

$$\sqrt{\left(\frac{\frac{2}{5}+\frac{3}{30}}{\frac{8}{3}}+1-\left(\frac{1}{2}\right)^{2}\right)\left(3+\frac{3}{4}\right)}$$

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

$$\sqrt{\left(\frac{\frac{2}{5}+\frac{1}{10}}{\frac{8}{3}}+1-\left(\frac{1}{2}\right)^{2}\right)\left(3+\frac{3}{4}\right)}$$

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

$$\sqrt{\left(\frac{\frac{4}{10}+\frac{1}{10}}{\frac{8}{3}}+1-\left(\frac{1}{2}\right)^{2}\right)\left(3+\frac{3}{4}\right)}$$

Since \frac{4}{10} and \frac{1}{10} have the same denominator, add them by adding their numerators.

$$\sqrt{\left(\frac{\frac{4+1}{10}}{\frac{8}{3}}+1-\left(\frac{1}{2}\right)^{2}\right)\left(3+\frac{3}{4}\right)}$$

Add 4 and 1 to get 5.

$$\sqrt{\left(\frac{\frac{5}{10}}{\frac{8}{3}}+1-\left(\frac{1}{2}\right)^{2}\right)\left(3+\frac{3}{4}\right)}$$

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

$$\sqrt{\left(\frac{\frac{1}{2}}{\frac{8}{3}}+1-\left(\frac{1}{2}\right)^{2}\right)\left(3+\frac{3}{4}\right)}$$

Divide \frac{1}{2} by \frac{8}{3} by multiplying \frac{1}{2} by the reciprocal of \frac{8}{3}.

$$\sqrt{\left(\frac{1}{2}\times \frac{3}{8}+1-\left(\frac{1}{2}\right)^{2}\right)\left(3+\frac{3}{4}\right)}$$

Multiply \frac{1}{2} times \frac{3}{8} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\left(\frac{1\times 3}{2\times 8}+1-\left(\frac{1}{2}\right)^{2}\right)\left(3+\frac{3}{4}\right)}$$

Do the multiplications in the fraction \frac{1\times 3}{2\times 8}.

$$\sqrt{\left(\frac{3}{16}+1-\left(\frac{1}{2}\right)^{2}\right)\left(3+\frac{3}{4}\right)}$$

Convert 1 to fraction \frac{16}{16}.

$$\sqrt{\left(\frac{3}{16}+\frac{16}{16}-\left(\frac{1}{2}\right)^{2}\right)\left(3+\frac{3}{4}\right)}$$

Since \frac{3}{16} and \frac{16}{16} have the same denominator, add them by adding their numerators.

$$\sqrt{\left(\frac{3+16}{16}-\left(\frac{1}{2}\right)^{2}\right)\left(3+\frac{3}{4}\right)}$$

Add 3 and 16 to get 19.

$$\sqrt{\left(\frac{19}{16}-\left(\frac{1}{2}\right)^{2}\right)\left(3+\frac{3}{4}\right)}$$

Calculate \frac{1}{2} to the power of 2 and get \frac{1}{4}.

$$\sqrt{\left(\frac{19}{16}-\frac{1}{4}\right)\left(3+\frac{3}{4}\right)}$$

Least common multiple of 16 and 4 is 16. Convert \frac{19}{16} and \frac{1}{4} to fractions with denominator 16.

$$\sqrt{\left(\frac{19}{16}-\frac{4}{16}\right)\left(3+\frac{3}{4}\right)}$$

Since \frac{19}{16} and \frac{4}{16} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{19-4}{16}\left(3+\frac{3}{4}\right)}$$

Subtract 4 from 19 to get 15.

$$\sqrt{\frac{15}{16}\left(3+\frac{3}{4}\right)}$$

Convert 3 to fraction \frac{12}{4}.

$$\sqrt{\frac{15}{16}\left(\frac{12}{4}+\frac{3}{4}\right)}$$

Since \frac{12}{4} and \frac{3}{4} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{15}{16}\times \frac{12+3}{4}}$$

Add 12 and 3 to get 15.

$$\sqrt{\frac{15}{16}\times \frac{15}{4}}$$

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

$$\sqrt{\frac{15\times 15}{16\times 4}}$$

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

$$\sqrt{\frac{225}{64}}$$

Rewrite the square root of the division \frac{225}{64} as the division of square roots \frac{\sqrt{225}}{\sqrt{64}}. Take the square root of both numerator and denominator.

$$\frac{15}{8}$$

Factor

$$\frac{3 \cdot 5}{2 ^ {3}} = 1\frac{7}{8} = 1.875$$