Solve sqrt{[(15/20-2/3):(4/6+5/3+frac{1}{3})+frac{5}{2}]*8}=

$$\sqrt { [ ( \frac { 15 } { 20 } - \frac { 2 } { 3 } ) : ( \frac { 4 } { 6 } + \frac { 5 } { 3 } + \frac { 1 } { 3 } ) + \frac { 5 } { 2 } ] \cdot 8 } =$$

Evaluate

$$\frac{9}{2}=4.5$$

Solution Steps

Reduce the fraction \frac{15}{20} to lowest terms by extracting and canceling out 5.

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

Least common multiple of 4 and 3 is 12. Convert \frac{3}{4} and \frac{2}{3} to fractions with denominator 12.

$$\sqrt{\left(\frac{\frac{9}{12}-\frac{8}{12}}{\frac{4}{6}+\frac{5}{3}+\frac{1}{3}}+\frac{5}{2}\right)\times 8}$$

Since \frac{9}{12} and \frac{8}{12} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\left(\frac{\frac{9-8}{12}}{\frac{4}{6}+\frac{5}{3}+\frac{1}{3}}+\frac{5}{2}\right)\times 8}$$

Subtract 8 from 9 to get 1.

$$\sqrt{\left(\frac{\frac{1}{12}}{\frac{4}{6}+\frac{5}{3}+\frac{1}{3}}+\frac{5}{2}\right)\times 8}$$

Reduce the fraction \frac{4}{6} to lowest terms by extracting and canceling out 2.

$$\sqrt{\left(\frac{\frac{1}{12}}{\frac{2}{3}+\frac{5}{3}+\frac{1}{3}}+\frac{5}{2}\right)\times 8}$$

Since \frac{2}{3} and \frac{5}{3} have the same denominator, add them by adding their numerators.

$$\sqrt{\left(\frac{\frac{1}{12}}{\frac{2+5}{3}+\frac{1}{3}}+\frac{5}{2}\right)\times 8}$$

Add 2 and 5 to get 7.

$$\sqrt{\left(\frac{\frac{1}{12}}{\frac{7}{3}+\frac{1}{3}}+\frac{5}{2}\right)\times 8}$$

Since \frac{7}{3} and \frac{1}{3} have the same denominator, add them by adding their numerators.

$$\sqrt{\left(\frac{\frac{1}{12}}{\frac{7+1}{3}}+\frac{5}{2}\right)\times 8}$$

Add 7 and 1 to get 8.

$$\sqrt{\left(\frac{\frac{1}{12}}{\frac{8}{3}}+\frac{5}{2}\right)\times 8}$$

Divide \frac{1}{12} by \frac{8}{3} by multiplying \frac{1}{12} by the reciprocal of \frac{8}{3}.

$$\sqrt{\left(\frac{1}{12}\times \frac{3}{8}+\frac{5}{2}\right)\times 8}$$

Multiply \frac{1}{12} times \frac{3}{8} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\left(\frac{1\times 3}{12\times 8}+\frac{5}{2}\right)\times 8}$$

Do the multiplications in the fraction \frac{1\times 3}{12\times 8}.

$$\sqrt{\left(\frac{3}{96}+\frac{5}{2}\right)\times 8}$$

Reduce the fraction \frac{3}{96} to lowest terms by extracting and canceling out 3.

$$\sqrt{\left(\frac{1}{32}+\frac{5}{2}\right)\times 8}$$

Least common multiple of 32 and 2 is 32. Convert \frac{1}{32} and \frac{5}{2} to fractions with denominator 32.

$$\sqrt{\left(\frac{1}{32}+\frac{80}{32}\right)\times 8}$$

Since \frac{1}{32} and \frac{80}{32} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{1+80}{32}\times 8}$$

Add 1 and 80 to get 81.

$$\sqrt{\frac{81}{32}\times 8}$$

Express \frac{81}{32}\times 8 as a single fraction.

$$\sqrt{\frac{81\times 8}{32}}$$

Multiply 81 and 8 to get 648.

$$\sqrt{\frac{648}{32}}$$

Reduce the fraction \frac{648}{32} to lowest terms by extracting and canceling out 8.

$$\sqrt{\frac{81}{4}}$$

Rewrite the square root of the division \frac{81}{4} as the division of square roots \frac{\sqrt{81}}{\sqrt{4}}. Take the square root of both numerator and denominator.

$$\frac{9}{2}$$

Factor

$$\frac{3 ^ {2}}{2} = 4\frac{1}{2} = 4.5$$