Solve sqrt{(5/2-7/6-3/4):(22/16-frac{10}{8})}]:sqrt{((frac{7}{15}+frac{3}{20}-frac{5}{24}):(frac{5}{4}-frac{9}{10})}]=

$$\sqrt { ( \frac { 5 } { 2 } - \frac { 7 } { 6 } - \frac { 3 } { 4 } ) : ( \frac { 22 } { 16 } - \frac { 10 } { 8 } ) } ] : \sqrt { ( ( \frac { 7 } { 15 } + \frac { 3 } { 20 } - \frac { 5 } { 24 } ) : ( \frac { 5 } { 4 } - \frac { 9 } { 10 } ) } ] =$$

Evaluate

$$\frac{\sqrt{42}}{3}\approx 2.160246899$$

Solution Steps

Least common multiple of 2 and 6 is 6. Convert \frac{5}{2} and \frac{7}{6} to fractions with denominator 6.

$$\sqrt{\frac{\frac{15}{6}-\frac{7}{6}-\frac{3}{4}}{\frac{22}{16}-\frac{10}{8}}}$$

Since \frac{15}{6} and \frac{7}{6} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{\frac{15-7}{6}-\frac{3}{4}}{\frac{22}{16}-\frac{10}{8}}}$$

Subtract 7 from 15 to get 8.

$$\sqrt{\frac{\frac{8}{6}-\frac{3}{4}}{\frac{22}{16}-\frac{10}{8}}}$$

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

$$\sqrt{\frac{\frac{4}{3}-\frac{3}{4}}{\frac{22}{16}-\frac{10}{8}}}$$

Least common multiple of 3 and 4 is 12. Convert \frac{4}{3} and \frac{3}{4} to fractions with denominator 12.

$$\sqrt{\frac{\frac{16}{12}-\frac{9}{12}}{\frac{22}{16}-\frac{10}{8}}}$$

Since \frac{16}{12} and \frac{9}{12} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{\frac{16-9}{12}}{\frac{22}{16}-\frac{10}{8}}}$$

Subtract 9 from 16 to get 7.

$$\sqrt{\frac{\frac{7}{12}}{\frac{22}{16}-\frac{10}{8}}}$$

Reduce the fraction \frac{22}{16} to lowest terms by extracting and canceling out 2.

$$\sqrt{\frac{\frac{7}{12}}{\frac{11}{8}-\frac{10}{8}}}$$

Since \frac{11}{8} and \frac{10}{8} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{\frac{7}{12}}{\frac{11-10}{8}}}$$

Subtract 10 from 11 to get 1.

$$\sqrt{\frac{\frac{7}{12}}{\frac{1}{8}}}$$

Divide \frac{7}{12} by \frac{1}{8} by multiplying \frac{7}{12} by the reciprocal of \frac{1}{8}.

$$\sqrt{\frac{7}{12}\times 8}$$

Express \frac{7}{12}\times 8 as a single fraction.

$$\sqrt{\frac{7\times 8}{12}}$$

Multiply 7 and 8 to get 56.

$$\sqrt{\frac{56}{12}}$$

Reduce the fraction \frac{56}{12} to lowest terms by extracting and canceling out 4.

$$\sqrt{\frac{14}{3}}$$

Rewrite the square root of the division \sqrt{\frac{14}{3}} as the division of square roots \frac{\sqrt{14}}{\sqrt{3}}.

$$\frac{\sqrt{14}}{\sqrt{3}}$$

Rationalize the denominator of \frac{\sqrt{14}}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{\sqrt{14}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{\sqrt{14}\sqrt{3}}{3}$$

To multiply \sqrt{14} and \sqrt{3}, multiply the numbers under the square root.

$$\frac{\sqrt{42}}{3}$$