Solve sqrt{(1/3*3/7+5/14)+(frac{3/2)^2*frac{5}{9}-frac{3}{8}*frac{4}{3}}{frac{9}{4}:frac{1}{5}+2-frac{3}{4}}}=

$$\sqrt { \frac { ( \frac { 1 } { 3 } \cdot \frac { 3 } { 7 } + \frac { 5 } { 14 } ) + ( \frac { 3 } { 2 } ) ^ { 2 } \cdot \frac { 5 } { 9 } - \frac { 3 } { 8 } \cdot \frac { 4 } { 3 } } { \frac { 9 } { 4 } : \frac { 1 } { 5 } + 2 - \frac { 3 } { 4 } } } =$$

Evaluate

$$\frac{\sqrt{10}}{10}\approx 0.316227766$$

Solution Steps

Multiply \frac{1}{3} times \frac{3}{7} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{\frac{1\times 3}{3\times 7}+\frac{5}{14}+\left(\frac{3}{2}\right)^{2}\times \frac{5}{9}-\frac{3}{8}\times \frac{4}{3}}{\frac{\frac{9}{4}}{\frac{1}{5}}+2-\frac{3}{4}}}$$

Cancel out 3 in both numerator and denominator.

$$\sqrt{\frac{\frac{1}{7}+\frac{5}{14}+\left(\frac{3}{2}\right)^{2}\times \frac{5}{9}-\frac{3}{8}\times \frac{4}{3}}{\frac{\frac{9}{4}}{\frac{1}{5}}+2-\frac{3}{4}}}$$

Least common multiple of 7 and 14 is 14. Convert \frac{1}{7} and \frac{5}{14} to fractions with denominator 14.

$$\sqrt{\frac{\frac{2}{14}+\frac{5}{14}+\left(\frac{3}{2}\right)^{2}\times \frac{5}{9}-\frac{3}{8}\times \frac{4}{3}}{\frac{\frac{9}{4}}{\frac{1}{5}}+2-\frac{3}{4}}}$$

Since \frac{2}{14} and \frac{5}{14} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{\frac{2+5}{14}+\left(\frac{3}{2}\right)^{2}\times \frac{5}{9}-\frac{3}{8}\times \frac{4}{3}}{\frac{\frac{9}{4}}{\frac{1}{5}}+2-\frac{3}{4}}}$$

Add 2 and 5 to get 7.

$$\sqrt{\frac{\frac{7}{14}+\left(\frac{3}{2}\right)^{2}\times \frac{5}{9}-\frac{3}{8}\times \frac{4}{3}}{\frac{\frac{9}{4}}{\frac{1}{5}}+2-\frac{3}{4}}}$$

Reduce the fraction \frac{7}{14} to lowest terms by extracting and canceling out 7.

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

Calculate \frac{3}{2} to the power of 2 and get \frac{9}{4}.

$$\sqrt{\frac{\frac{1}{2}+\frac{9}{4}\times \frac{5}{9}-\frac{3}{8}\times \frac{4}{3}}{\frac{\frac{9}{4}}{\frac{1}{5}}+2-\frac{3}{4}}}$$

Multiply \frac{9}{4} times \frac{5}{9} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{\frac{1}{2}+\frac{9\times 5}{4\times 9}-\frac{3}{8}\times \frac{4}{3}}{\frac{\frac{9}{4}}{\frac{1}{5}}+2-\frac{3}{4}}}$$

Cancel out 9 in both numerator and denominator.

$$\sqrt{\frac{\frac{1}{2}+\frac{5}{4}-\frac{3}{8}\times \frac{4}{3}}{\frac{\frac{9}{4}}{\frac{1}{5}}+2-\frac{3}{4}}}$$

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

$$\sqrt{\frac{\frac{2}{4}+\frac{5}{4}-\frac{3}{8}\times \frac{4}{3}}{\frac{\frac{9}{4}}{\frac{1}{5}}+2-\frac{3}{4}}}$$

Since \frac{2}{4} and \frac{5}{4} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{\frac{2+5}{4}-\frac{3}{8}\times \frac{4}{3}}{\frac{\frac{9}{4}}{\frac{1}{5}}+2-\frac{3}{4}}}$$

Add 2 and 5 to get 7.

$$\sqrt{\frac{\frac{7}{4}-\frac{3}{8}\times \frac{4}{3}}{\frac{\frac{9}{4}}{\frac{1}{5}}+2-\frac{3}{4}}}$$

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

$$\sqrt{\frac{\frac{7}{4}-\frac{3\times 4}{8\times 3}}{\frac{\frac{9}{4}}{\frac{1}{5}}+2-\frac{3}{4}}}$$

Cancel out 3 in both numerator and denominator.

$$\sqrt{\frac{\frac{7}{4}-\frac{4}{8}}{\frac{\frac{9}{4}}{\frac{1}{5}}+2-\frac{3}{4}}}$$

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

$$\sqrt{\frac{\frac{7}{4}-\frac{1}{2}}{\frac{\frac{9}{4}}{\frac{1}{5}}+2-\frac{3}{4}}}$$

Least common multiple of 4 and 2 is 4. Convert \frac{7}{4} and \frac{1}{2} to fractions with denominator 4.

$$\sqrt{\frac{\frac{7}{4}-\frac{2}{4}}{\frac{\frac{9}{4}}{\frac{1}{5}}+2-\frac{3}{4}}}$$

Since \frac{7}{4} and \frac{2}{4} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{\frac{7-2}{4}}{\frac{\frac{9}{4}}{\frac{1}{5}}+2-\frac{3}{4}}}$$

Subtract 2 from 7 to get 5.

$$\sqrt{\frac{\frac{5}{4}}{\frac{\frac{9}{4}}{\frac{1}{5}}+2-\frac{3}{4}}}$$

Divide \frac{9}{4} by \frac{1}{5} by multiplying \frac{9}{4} by the reciprocal of \frac{1}{5}.

$$\sqrt{\frac{\frac{5}{4}}{\frac{9}{4}\times 5+2-\frac{3}{4}}}$$

Express \frac{9}{4}\times 5 as a single fraction.

$$\sqrt{\frac{\frac{5}{4}}{\frac{9\times 5}{4}+2-\frac{3}{4}}}$$

Multiply 9 and 5 to get 45.

$$\sqrt{\frac{\frac{5}{4}}{\frac{45}{4}+2-\frac{3}{4}}}$$

Convert 2 to fraction \frac{8}{4}.

$$\sqrt{\frac{\frac{5}{4}}{\frac{45}{4}+\frac{8}{4}-\frac{3}{4}}}$$

Since \frac{45}{4} and \frac{8}{4} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{\frac{5}{4}}{\frac{45+8}{4}-\frac{3}{4}}}$$

Add 45 and 8 to get 53.

$$\sqrt{\frac{\frac{5}{4}}{\frac{53}{4}-\frac{3}{4}}}$$

Since \frac{53}{4} and \frac{3}{4} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{\frac{5}{4}}{\frac{53-3}{4}}}$$

Subtract 3 from 53 to get 50.

$$\sqrt{\frac{\frac{5}{4}}{\frac{50}{4}}}$$

Reduce the fraction \frac{50}{4} to lowest terms by extracting and canceling out 2.

$$\sqrt{\frac{\frac{5}{4}}{\frac{25}{2}}}$$

Divide \frac{5}{4} by \frac{25}{2} by multiplying \frac{5}{4} by the reciprocal of \frac{25}{2}.

$$\sqrt{\frac{5}{4}\times \frac{2}{25}}$$

Multiply \frac{5}{4} times \frac{2}{25} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{5\times 2}{4\times 25}}$$

Do the multiplications in the fraction \frac{5\times 2}{4\times 25}.

$$\sqrt{\frac{10}{100}}$$

Reduce the fraction \frac{10}{100} to lowest terms by extracting and canceling out 10.

$$\sqrt{\frac{1}{10}}$$

Rewrite the square root of the division \sqrt{\frac{1}{10}} as the division of square roots \frac{\sqrt{1}}{\sqrt{10}}.

$$\frac{\sqrt{1}}{\sqrt{10}}$$

Calculate the square root of 1 and get 1.

$$\frac{1}{\sqrt{10}}$$

Rationalize the denominator of \frac{1}{\sqrt{10}} by multiplying numerator and denominator by \sqrt{10}.

$$\frac{\sqrt{10}}{\left(\sqrt{10}\right)^{2}}$$

The square of \sqrt{10} is 10.

$$\frac{\sqrt{10}}{10}$$