Solve sqrt{1+1/2}{1+1/2[(9/2-1+3/2)*(1+frac{7}{5}-frac{6}{5})+frac{3}{2}*frac{9}{4}]}*frac{5}{2}+frac{31}{12}

$$\sqrt { 1 + \frac { 1 } { 2 } } \{ 1 + \frac { 1 } { 2 } [ ( \frac { 9 } { 2 } - 1 + \frac { 3 } { 2 } ) \cdot ( 1 + \frac { 7 } { 5 } - \frac { 6 } { 5 } ) + \frac { 3 } { 2 } \cdot \frac { 9 } { 4 } ] \} \cdot \frac { 5 } { 2 } + \frac { 31 } { 12 }$$

Evaluate

$$\frac{455\sqrt{6}}{64}+\frac{31}{12}\approx 19.997674473$$

Short Solution Steps

Convert 1 to fraction \frac{2}{2}.

$$\sqrt{\frac{2}{2}+\frac{1}{2}}\left(1+\frac{1}{2}\left(\left(\frac{9}{2}-1+\frac{3}{2}\right)\left(1+\frac{7}{5}-\frac{6}{5}\right)+\frac{3}{2}\times \frac{9}{4}\right)\right)\times \frac{5}{2}+\frac{31}{12}$$

Since \frac{2}{2} and \frac{1}{2} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{2+1}{2}}\left(1+\frac{1}{2}\left(\left(\frac{9}{2}-1+\frac{3}{2}\right)\left(1+\frac{7}{5}-\frac{6}{5}\right)+\frac{3}{2}\times \frac{9}{4}\right)\right)\times \frac{5}{2}+\frac{31}{12}$$

Add 2 and 1 to get 3.

$$\sqrt{\frac{3}{2}}\left(1+\frac{1}{2}\left(\left(\frac{9}{2}-1+\frac{3}{2}\right)\left(1+\frac{7}{5}-\frac{6}{5}\right)+\frac{3}{2}\times \frac{9}{4}\right)\right)\times \frac{5}{2}+\frac{31}{12}$$

Rewrite the square root of the division \sqrt{\frac{3}{2}} as the division of square roots \frac{\sqrt{3}}{\sqrt{2}}.

$$\frac{\sqrt{3}}{\sqrt{2}}\left(1+\frac{1}{2}\left(\left(\frac{9}{2}-1+\frac{3}{2}\right)\left(1+\frac{7}{5}-\frac{6}{5}\right)+\frac{3}{2}\times \frac{9}{4}\right)\right)\times \frac{5}{2}+\frac{31}{12}$$

Rationalize the denominator of \frac{\sqrt{3}}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{\sqrt{3}\sqrt{2}}{\left(\sqrt{2}\right)^{2}}\left(1+\frac{1}{2}\left(\left(\frac{9}{2}-1+\frac{3}{2}\right)\left(1+\frac{7}{5}-\frac{6}{5}\right)+\frac{3}{2}\times \frac{9}{4}\right)\right)\times \frac{5}{2}+\frac{31}{12}$$

The square of \sqrt{2} is 2.

$$\frac{\sqrt{3}\sqrt{2}}{2}\left(1+\frac{1}{2}\left(\left(\frac{9}{2}-1+\frac{3}{2}\right)\left(1+\frac{7}{5}-\frac{6}{5}\right)+\frac{3}{2}\times \frac{9}{4}\right)\right)\times \frac{5}{2}+\frac{31}{12}$$

To multiply \sqrt{3} and \sqrt{2}, multiply the numbers under the square root.

$$\frac{\sqrt{6}}{2}\left(1+\frac{1}{2}\left(\left(\frac{9}{2}-1+\frac{3}{2}\right)\left(1+\frac{7}{5}-\frac{6}{5}\right)+\frac{3}{2}\times \frac{9}{4}\right)\right)\times \frac{5}{2}+\frac{31}{12}$$

Convert 1 to fraction \frac{2}{2}.

$$\frac{\sqrt{6}}{2}\left(1+\frac{1}{2}\left(\left(\frac{9}{2}-\frac{2}{2}+\frac{3}{2}\right)\left(1+\frac{7}{5}-\frac{6}{5}\right)+\frac{3}{2}\times \frac{9}{4}\right)\right)\times \frac{5}{2}+\frac{31}{12}$$

Since \frac{9}{2} and \frac{2}{2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\sqrt{6}}{2}\left(1+\frac{1}{2}\left(\left(\frac{9-2}{2}+\frac{3}{2}\right)\left(1+\frac{7}{5}-\frac{6}{5}\right)+\frac{3}{2}\times \frac{9}{4}\right)\right)\times \frac{5}{2}+\frac{31}{12}$$

Subtract 2 from 9 to get 7.

$$\frac{\sqrt{6}}{2}\left(1+\frac{1}{2}\left(\left(\frac{7}{2}+\frac{3}{2}\right)\left(1+\frac{7}{5}-\frac{6}{5}\right)+\frac{3}{2}\times \frac{9}{4}\right)\right)\times \frac{5}{2}+\frac{31}{12}$$

Since \frac{7}{2} and \frac{3}{2} have the same denominator, add them by adding their numerators.

$$\frac{\sqrt{6}}{2}\left(1+\frac{1}{2}\left(\frac{7+3}{2}\left(1+\frac{7}{5}-\frac{6}{5}\right)+\frac{3}{2}\times \frac{9}{4}\right)\right)\times \frac{5}{2}+\frac{31}{12}$$

Add 7 and 3 to get 10.

$$\frac{\sqrt{6}}{2}\left(1+\frac{1}{2}\left(\frac{10}{2}\left(1+\frac{7}{5}-\frac{6}{5}\right)+\frac{3}{2}\times \frac{9}{4}\right)\right)\times \frac{5}{2}+\frac{31}{12}$$

Divide 10 by 2 to get 5.

$$\frac{\sqrt{6}}{2}\left(1+\frac{1}{2}\left(5\left(1+\frac{7}{5}-\frac{6}{5}\right)+\frac{3}{2}\times \frac{9}{4}\right)\right)\times \frac{5}{2}+\frac{31}{12}$$

Convert 1 to fraction \frac{5}{5}.

$$\frac{\sqrt{6}}{2}\left(1+\frac{1}{2}\left(5\left(\frac{5}{5}+\frac{7}{5}-\frac{6}{5}\right)+\frac{3}{2}\times \frac{9}{4}\right)\right)\times \frac{5}{2}+\frac{31}{12}$$

Since \frac{5}{5} and \frac{7}{5} have the same denominator, add them by adding their numerators.

$$\frac{\sqrt{6}}{2}\left(1+\frac{1}{2}\left(5\left(\frac{5+7}{5}-\frac{6}{5}\right)+\frac{3}{2}\times \frac{9}{4}\right)\right)\times \frac{5}{2}+\frac{31}{12}$$

Add 5 and 7 to get 12.

$$\frac{\sqrt{6}}{2}\left(1+\frac{1}{2}\left(5\left(\frac{12}{5}-\frac{6}{5}\right)+\frac{3}{2}\times \frac{9}{4}\right)\right)\times \frac{5}{2}+\frac{31}{12}$$

Since \frac{12}{5} and \frac{6}{5} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\sqrt{6}}{2}\left(1+\frac{1}{2}\left(5\times \frac{12-6}{5}+\frac{3}{2}\times \frac{9}{4}\right)\right)\times \frac{5}{2}+\frac{31}{12}$$

Subtract 6 from 12 to get 6.

$$\frac{\sqrt{6}}{2}\left(1+\frac{1}{2}\left(5\times \frac{6}{5}+\frac{3}{2}\times \frac{9}{4}\right)\right)\times \frac{5}{2}+\frac{31}{12}$$

Cancel out 5 and 5.

$$\frac{\sqrt{6}}{2}\left(1+\frac{1}{2}\left(6+\frac{3}{2}\times \frac{9}{4}\right)\right)\times \frac{5}{2}+\frac{31}{12}$$

Multiply \frac{3}{2} times \frac{9}{4} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\sqrt{6}}{2}\left(1+\frac{1}{2}\left(6+\frac{3\times 9}{2\times 4}\right)\right)\times \frac{5}{2}+\frac{31}{12}$$

Do the multiplications in the fraction \frac{3\times 9}{2\times 4}.

$$\frac{\sqrt{6}}{2}\left(1+\frac{1}{2}\left(6+\frac{27}{8}\right)\right)\times \frac{5}{2}+\frac{31}{12}$$

Convert 6 to fraction \frac{48}{8}.

$$\frac{\sqrt{6}}{2}\left(1+\frac{1}{2}\left(\frac{48}{8}+\frac{27}{8}\right)\right)\times \frac{5}{2}+\frac{31}{12}$$

Since \frac{48}{8} and \frac{27}{8} have the same denominator, add them by adding their numerators.

$$\frac{\sqrt{6}}{2}\left(1+\frac{1}{2}\times \frac{48+27}{8}\right)\times \frac{5}{2}+\frac{31}{12}$$

Add 48 and 27 to get 75.

$$\frac{\sqrt{6}}{2}\left(1+\frac{1}{2}\times \frac{75}{8}\right)\times \frac{5}{2}+\frac{31}{12}$$

Multiply \frac{1}{2} times \frac{75}{8} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\sqrt{6}}{2}\left(1+\frac{1\times 75}{2\times 8}\right)\times \frac{5}{2}+\frac{31}{12}$$

Do the multiplications in the fraction \frac{1\times 75}{2\times 8}.

$$\frac{\sqrt{6}}{2}\left(1+\frac{75}{16}\right)\times \frac{5}{2}+\frac{31}{12}$$

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

$$\frac{\sqrt{6}}{2}\left(\frac{16}{16}+\frac{75}{16}\right)\times \frac{5}{2}+\frac{31}{12}$$

Since \frac{16}{16} and \frac{75}{16} have the same denominator, add them by adding their numerators.

$$\frac{\sqrt{6}}{2}\times \frac{16+75}{16}\times \frac{5}{2}+\frac{31}{12}$$

Add 16 and 75 to get 91.

$$\frac{\sqrt{6}}{2}\times \frac{91}{16}\times \frac{5}{2}+\frac{31}{12}$$

Multiply \frac{91}{16} times \frac{5}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\sqrt{6}}{2}\times \frac{91\times 5}{16\times 2}+\frac{31}{12}$$

Do the multiplications in the fraction \frac{91\times 5}{16\times 2}.

$$\frac{\sqrt{6}}{2}\times \frac{455}{32}+\frac{31}{12}$$

Multiply \frac{\sqrt{6}}{2} times \frac{455}{32} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\sqrt{6}\times 455}{2\times 32}+\frac{31}{12}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2\times 32 and 12 is 192. Multiply \frac{\sqrt{6}\times 455}{2\times 32} times \frac{3}{3}. Multiply \frac{31}{12} times \frac{16}{16}.

$$\frac{3\sqrt{6}\times 455}{192}+\frac{31\times 16}{192}$$

Since \frac{3\sqrt{6}\times 455}{192} and \frac{31\times 16}{192} have the same denominator, add them by adding their numerators.

$$\frac{3\sqrt{6}\times 455+31\times 16}{192}$$

Do the multiplications in 3\sqrt{6}\times 455+31\times 16.

$$\frac{1365\sqrt{6}+496}{192}$$

Factor

$$\frac{1365 \sqrt{6} + 496}{192} = 19.99767447343249$$