Solve sqrt{6/5^2*{[(3/2+2/3+2)*3-1/4]:(frac{7}{3}-frac{1}{2}*frac{7}{5})-frac{3}{2}}}

$$\sqrt { \frac { 6 } { 5 ^ { 2 } } \times \{ [ ( \frac { 3 } { 2 } + \frac { 2 } { 3 } + 2 ) \times 3 - \frac { 1 } { 4 } ] : ( \frac { 7 } { 3 } - \frac { 1 } { 2 } \times \frac { 7 } { 5 } ) - \frac { 3 } { 2 } \} }$$

Evaluate

$$\frac{6}{5}=1.2$$

Solution Steps

Calculate 5 to the power of 2 and get 25.

$$\sqrt{\frac{6}{25}\left(\frac{\left(\frac{3}{2}+\frac{2}{3}+2\right)\times 3-\frac{1}{4}}{\frac{7}{3}-\frac{1}{2}\times \frac{7}{5}}-\frac{3}{2}\right)}$$

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

$$\sqrt{\frac{6}{25}\left(\frac{\left(\frac{9}{6}+\frac{4}{6}+2\right)\times 3-\frac{1}{4}}{\frac{7}{3}-\frac{1}{2}\times \frac{7}{5}}-\frac{3}{2}\right)}$$

Since \frac{9}{6} and \frac{4}{6} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{6}{25}\left(\frac{\left(\frac{9+4}{6}+2\right)\times 3-\frac{1}{4}}{\frac{7}{3}-\frac{1}{2}\times \frac{7}{5}}-\frac{3}{2}\right)}$$

Add 9 and 4 to get 13.

$$\sqrt{\frac{6}{25}\left(\frac{\left(\frac{13}{6}+2\right)\times 3-\frac{1}{4}}{\frac{7}{3}-\frac{1}{2}\times \frac{7}{5}}-\frac{3}{2}\right)}$$

Convert 2 to fraction \frac{12}{6}.

$$\sqrt{\frac{6}{25}\left(\frac{\left(\frac{13}{6}+\frac{12}{6}\right)\times 3-\frac{1}{4}}{\frac{7}{3}-\frac{1}{2}\times \frac{7}{5}}-\frac{3}{2}\right)}$$

Since \frac{13}{6} and \frac{12}{6} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{6}{25}\left(\frac{\frac{13+12}{6}\times 3-\frac{1}{4}}{\frac{7}{3}-\frac{1}{2}\times \frac{7}{5}}-\frac{3}{2}\right)}$$

Add 13 and 12 to get 25.

$$\sqrt{\frac{6}{25}\left(\frac{\frac{25}{6}\times 3-\frac{1}{4}}{\frac{7}{3}-\frac{1}{2}\times \frac{7}{5}}-\frac{3}{2}\right)}$$

Express \frac{25}{6}\times 3 as a single fraction.

$$\sqrt{\frac{6}{25}\left(\frac{\frac{25\times 3}{6}-\frac{1}{4}}{\frac{7}{3}-\frac{1}{2}\times \frac{7}{5}}-\frac{3}{2}\right)}$$

Multiply 25 and 3 to get 75.

$$\sqrt{\frac{6}{25}\left(\frac{\frac{75}{6}-\frac{1}{4}}{\frac{7}{3}-\frac{1}{2}\times \frac{7}{5}}-\frac{3}{2}\right)}$$

Reduce the fraction \frac{75}{6} to lowest terms by extracting and canceling out 3.

$$\sqrt{\frac{6}{25}\left(\frac{\frac{25}{2}-\frac{1}{4}}{\frac{7}{3}-\frac{1}{2}\times \frac{7}{5}}-\frac{3}{2}\right)}$$

Least common multiple of 2 and 4 is 4. Convert \frac{25}{2} and \frac{1}{4} to fractions with denominator 4.

$$\sqrt{\frac{6}{25}\left(\frac{\frac{50}{4}-\frac{1}{4}}{\frac{7}{3}-\frac{1}{2}\times \frac{7}{5}}-\frac{3}{2}\right)}$$

Since \frac{50}{4} and \frac{1}{4} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{6}{25}\left(\frac{\frac{50-1}{4}}{\frac{7}{3}-\frac{1}{2}\times \frac{7}{5}}-\frac{3}{2}\right)}$$

Subtract 1 from 50 to get 49.

$$\sqrt{\frac{6}{25}\left(\frac{\frac{49}{4}}{\frac{7}{3}-\frac{1}{2}\times \frac{7}{5}}-\frac{3}{2}\right)}$$

Multiply \frac{1}{2} times \frac{7}{5} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{6}{25}\left(\frac{\frac{49}{4}}{\frac{7}{3}-\frac{1\times 7}{2\times 5}}-\frac{3}{2}\right)}$$

Do the multiplications in the fraction \frac{1\times 7}{2\times 5}.

$$\sqrt{\frac{6}{25}\left(\frac{\frac{49}{4}}{\frac{7}{3}-\frac{7}{10}}-\frac{3}{2}\right)}$$

Least common multiple of 3 and 10 is 30. Convert \frac{7}{3} and \frac{7}{10} to fractions with denominator 30.

$$\sqrt{\frac{6}{25}\left(\frac{\frac{49}{4}}{\frac{70}{30}-\frac{21}{30}}-\frac{3}{2}\right)}$$

Since \frac{70}{30} and \frac{21}{30} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{6}{25}\left(\frac{\frac{49}{4}}{\frac{70-21}{30}}-\frac{3}{2}\right)}$$

Subtract 21 from 70 to get 49.

$$\sqrt{\frac{6}{25}\left(\frac{\frac{49}{4}}{\frac{49}{30}}-\frac{3}{2}\right)}$$

Divide \frac{49}{4} by \frac{49}{30} by multiplying \frac{49}{4} by the reciprocal of \frac{49}{30}.

$$\sqrt{\frac{6}{25}\left(\frac{49}{4}\times \frac{30}{49}-\frac{3}{2}\right)}$$

Multiply \frac{49}{4} times \frac{30}{49} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{6}{25}\left(\frac{49\times 30}{4\times 49}-\frac{3}{2}\right)}$$

Cancel out 49 in both numerator and denominator.

$$\sqrt{\frac{6}{25}\left(\frac{30}{4}-\frac{3}{2}\right)}$$

Reduce the fraction \frac{30}{4} to lowest terms by extracting and canceling out 2.

$$\sqrt{\frac{6}{25}\left(\frac{15}{2}-\frac{3}{2}\right)}$$

Since \frac{15}{2} and \frac{3}{2} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{6}{25}\times \frac{15-3}{2}}$$

Subtract 3 from 15 to get 12.

$$\sqrt{\frac{6}{25}\times \frac{12}{2}}$$

Divide 12 by 2 to get 6.

$$\sqrt{\frac{6}{25}\times 6}$$

Express \frac{6}{25}\times 6 as a single fraction.

$$\sqrt{\frac{6\times 6}{25}}$$

Multiply 6 and 6 to get 36.

$$\sqrt{\frac{36}{25}}$$

Rewrite the square root of the division \frac{36}{25} as the division of square roots \frac{\sqrt{36}}{\sqrt{25}}. Take the square root of both numerator and denominator.

$$\frac{6}{5}$$

Factor

$$\frac{2 \cdot 3}{5} = 1\frac{1}{5} = 1.2$$