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

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

Evaluate

$$\frac{\sqrt{1246}}{28}\approx 1.260668757$$

Solution Steps

Add 2 and \frac{3}{5} to get \frac{13}{5}.

$$\frac{\sqrt{\frac{\frac{\left(\frac{13}{5}\right)^{2}}{\left(1+\frac{8}{5}\right)^{2}}+\frac{7}{8}\left(1+\frac{3}{4}\right)+\frac{5}{14}\left(\frac{1}{2}+\frac{1}{5}\right)}{2-\frac{1}{4}}}}{\sqrt{\frac{\left(4-\frac{3}{2}\right)^{3}}{\left(3-\frac{1}{2}\right)^{2}}-\left(\frac{8}{3}-\frac{5}{2}+1+\frac{1}{3}-\frac{1}{2}\right)\left(1+\frac{1}{2}\right)}}$$

Calculate \frac{13}{5} to the power of 2 and get \frac{169}{25}.

$$\frac{\sqrt{\frac{\frac{\frac{169}{25}}{\left(1+\frac{8}{5}\right)^{2}}+\frac{7}{8}\left(1+\frac{3}{4}\right)+\frac{5}{14}\left(\frac{1}{2}+\frac{1}{5}\right)}{2-\frac{1}{4}}}}{\sqrt{\frac{\left(4-\frac{3}{2}\right)^{3}}{\left(3-\frac{1}{2}\right)^{2}}-\left(\frac{8}{3}-\frac{5}{2}+1+\frac{1}{3}-\frac{1}{2}\right)\left(1+\frac{1}{2}\right)}}$$

Add 1 and \frac{8}{5} to get \frac{13}{5}.

$$\frac{\sqrt{\frac{\frac{\frac{169}{25}}{\left(\frac{13}{5}\right)^{2}}+\frac{7}{8}\left(1+\frac{3}{4}\right)+\frac{5}{14}\left(\frac{1}{2}+\frac{1}{5}\right)}{2-\frac{1}{4}}}}{\sqrt{\frac{\left(4-\frac{3}{2}\right)^{3}}{\left(3-\frac{1}{2}\right)^{2}}-\left(\frac{8}{3}-\frac{5}{2}+1+\frac{1}{3}-\frac{1}{2}\right)\left(1+\frac{1}{2}\right)}}$$

Calculate \frac{13}{5} to the power of 2 and get \frac{169}{25}.

$$\frac{\sqrt{\frac{\frac{\frac{169}{25}}{\frac{169}{25}}+\frac{7}{8}\left(1+\frac{3}{4}\right)+\frac{5}{14}\left(\frac{1}{2}+\frac{1}{5}\right)}{2-\frac{1}{4}}}}{\sqrt{\frac{\left(4-\frac{3}{2}\right)^{3}}{\left(3-\frac{1}{2}\right)^{2}}-\left(\frac{8}{3}-\frac{5}{2}+1+\frac{1}{3}-\frac{1}{2}\right)\left(1+\frac{1}{2}\right)}}$$

Divide \frac{169}{25} by \frac{169}{25} to get 1.

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

Add 1 and \frac{3}{4} to get \frac{7}{4}.

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

Multiply \frac{7}{8} and \frac{7}{4} to get \frac{49}{32}.

$$\frac{\sqrt{\frac{1+\frac{49}{32}+\frac{5}{14}\left(\frac{1}{2}+\frac{1}{5}\right)}{2-\frac{1}{4}}}}{\sqrt{\frac{\left(4-\frac{3}{2}\right)^{3}}{\left(3-\frac{1}{2}\right)^{2}}-\left(\frac{8}{3}-\frac{5}{2}+1+\frac{1}{3}-\frac{1}{2}\right)\left(1+\frac{1}{2}\right)}}$$

Add 1 and \frac{49}{32} to get \frac{81}{32}.

$$\frac{\sqrt{\frac{\frac{81}{32}+\frac{5}{14}\left(\frac{1}{2}+\frac{1}{5}\right)}{2-\frac{1}{4}}}}{\sqrt{\frac{\left(4-\frac{3}{2}\right)^{3}}{\left(3-\frac{1}{2}\right)^{2}}-\left(\frac{8}{3}-\frac{5}{2}+1+\frac{1}{3}-\frac{1}{2}\right)\left(1+\frac{1}{2}\right)}}$$

Add \frac{1}{2} and \frac{1}{5} to get \frac{7}{10}.

$$\frac{\sqrt{\frac{\frac{81}{32}+\frac{5}{14}\times \frac{7}{10}}{2-\frac{1}{4}}}}{\sqrt{\frac{\left(4-\frac{3}{2}\right)^{3}}{\left(3-\frac{1}{2}\right)^{2}}-\left(\frac{8}{3}-\frac{5}{2}+1+\frac{1}{3}-\frac{1}{2}\right)\left(1+\frac{1}{2}\right)}}$$

Multiply \frac{5}{14} and \frac{7}{10} to get \frac{1}{4}.

$$\frac{\sqrt{\frac{\frac{81}{32}+\frac{1}{4}}{2-\frac{1}{4}}}}{\sqrt{\frac{\left(4-\frac{3}{2}\right)^{3}}{\left(3-\frac{1}{2}\right)^{2}}-\left(\frac{8}{3}-\frac{5}{2}+1+\frac{1}{3}-\frac{1}{2}\right)\left(1+\frac{1}{2}\right)}}$$

Add \frac{81}{32} and \frac{1}{4} to get \frac{89}{32}.

$$\frac{\sqrt{\frac{\frac{89}{32}}{2-\frac{1}{4}}}}{\sqrt{\frac{\left(4-\frac{3}{2}\right)^{3}}{\left(3-\frac{1}{2}\right)^{2}}-\left(\frac{8}{3}-\frac{5}{2}+1+\frac{1}{3}-\frac{1}{2}\right)\left(1+\frac{1}{2}\right)}}$$

Subtract \frac{1}{4} from 2 to get \frac{7}{4}.

$$\frac{\sqrt{\frac{\frac{89}{32}}{\frac{7}{4}}}}{\sqrt{\frac{\left(4-\frac{3}{2}\right)^{3}}{\left(3-\frac{1}{2}\right)^{2}}-\left(\frac{8}{3}-\frac{5}{2}+1+\frac{1}{3}-\frac{1}{2}\right)\left(1+\frac{1}{2}\right)}}$$

Divide \frac{89}{32} by \frac{7}{4} by multiplying \frac{89}{32} by the reciprocal of \frac{7}{4}.

$$\frac{\sqrt{\frac{89}{32}\times \frac{4}{7}}}{\sqrt{\frac{\left(4-\frac{3}{2}\right)^{3}}{\left(3-\frac{1}{2}\right)^{2}}-\left(\frac{8}{3}-\frac{5}{2}+1+\frac{1}{3}-\frac{1}{2}\right)\left(1+\frac{1}{2}\right)}}$$

Multiply \frac{89}{32} and \frac{4}{7} to get \frac{89}{56}.

$$\frac{\sqrt{\frac{89}{56}}}{\sqrt{\frac{\left(4-\frac{3}{2}\right)^{3}}{\left(3-\frac{1}{2}\right)^{2}}-\left(\frac{8}{3}-\frac{5}{2}+1+\frac{1}{3}-\frac{1}{2}\right)\left(1+\frac{1}{2}\right)}}$$

Rewrite the square root of the division \sqrt{\frac{89}{56}} as the division of square roots \frac{\sqrt{89}}{\sqrt{56}}.

$$\frac{\frac{\sqrt{89}}{\sqrt{56}}}{\sqrt{\frac{\left(4-\frac{3}{2}\right)^{3}}{\left(3-\frac{1}{2}\right)^{2}}-\left(\frac{8}{3}-\frac{5}{2}+1+\frac{1}{3}-\frac{1}{2}\right)\left(1+\frac{1}{2}\right)}}$$

Factor 56=2^{2}\times 14. Rewrite the square root of the product \sqrt{2^{2}\times 14} as the product of square roots \sqrt{2^{2}}\sqrt{14}. Take the square root of 2^{2}.

$$\frac{\frac{\sqrt{89}}{2\sqrt{14}}}{\sqrt{\frac{\left(4-\frac{3}{2}\right)^{3}}{\left(3-\frac{1}{2}\right)^{2}}-\left(\frac{8}{3}-\frac{5}{2}+1+\frac{1}{3}-\frac{1}{2}\right)\left(1+\frac{1}{2}\right)}}$$

Rationalize the denominator of \frac{\sqrt{89}}{2\sqrt{14}} by multiplying numerator and denominator by \sqrt{14}.

$$\frac{\frac{\sqrt{89}\sqrt{14}}{2\left(\sqrt{14}\right)^{2}}}{\sqrt{\frac{\left(4-\frac{3}{2}\right)^{3}}{\left(3-\frac{1}{2}\right)^{2}}-\left(\frac{8}{3}-\frac{5}{2}+1+\frac{1}{3}-\frac{1}{2}\right)\left(1+\frac{1}{2}\right)}}$$

The square of \sqrt{14} is 14.

$$\frac{\frac{\sqrt{89}\sqrt{14}}{2\times 14}}{\sqrt{\frac{\left(4-\frac{3}{2}\right)^{3}}{\left(3-\frac{1}{2}\right)^{2}}-\left(\frac{8}{3}-\frac{5}{2}+1+\frac{1}{3}-\frac{1}{2}\right)\left(1+\frac{1}{2}\right)}}$$

To multiply \sqrt{89} and \sqrt{14}, multiply the numbers under the square root.

$$\frac{\frac{\sqrt{1246}}{2\times 14}}{\sqrt{\frac{\left(4-\frac{3}{2}\right)^{3}}{\left(3-\frac{1}{2}\right)^{2}}-\left(\frac{8}{3}-\frac{5}{2}+1+\frac{1}{3}-\frac{1}{2}\right)\left(1+\frac{1}{2}\right)}}$$

Multiply 2 and 14 to get 28.

$$\frac{\frac{\sqrt{1246}}{28}}{\sqrt{\frac{\left(4-\frac{3}{2}\right)^{3}}{\left(3-\frac{1}{2}\right)^{2}}-\left(\frac{8}{3}-\frac{5}{2}+1+\frac{1}{3}-\frac{1}{2}\right)\left(1+\frac{1}{2}\right)}}$$

Subtract \frac{3}{2} from 4 to get \frac{5}{2}.

$$\frac{\frac{\sqrt{1246}}{28}}{\sqrt{\frac{\left(\frac{5}{2}\right)^{3}}{\left(3-\frac{1}{2}\right)^{2}}-\left(\frac{8}{3}-\frac{5}{2}+1+\frac{1}{3}-\frac{1}{2}\right)\left(1+\frac{1}{2}\right)}}$$

Calculate \frac{5}{2} to the power of 3 and get \frac{125}{8}.

$$\frac{\frac{\sqrt{1246}}{28}}{\sqrt{\frac{\frac{125}{8}}{\left(3-\frac{1}{2}\right)^{2}}-\left(\frac{8}{3}-\frac{5}{2}+1+\frac{1}{3}-\frac{1}{2}\right)\left(1+\frac{1}{2}\right)}}$$

Subtract \frac{1}{2} from 3 to get \frac{5}{2}.

$$\frac{\frac{\sqrt{1246}}{28}}{\sqrt{\frac{\frac{125}{8}}{\left(\frac{5}{2}\right)^{2}}-\left(\frac{8}{3}-\frac{5}{2}+1+\frac{1}{3}-\frac{1}{2}\right)\left(1+\frac{1}{2}\right)}}$$

Calculate \frac{5}{2} to the power of 2 and get \frac{25}{4}.

$$\frac{\frac{\sqrt{1246}}{28}}{\sqrt{\frac{\frac{125}{8}}{\frac{25}{4}}-\left(\frac{8}{3}-\frac{5}{2}+1+\frac{1}{3}-\frac{1}{2}\right)\left(1+\frac{1}{2}\right)}}$$

Divide \frac{125}{8} by \frac{25}{4} by multiplying \frac{125}{8} by the reciprocal of \frac{25}{4}.

$$\frac{\frac{\sqrt{1246}}{28}}{\sqrt{\frac{125}{8}\times \frac{4}{25}-\left(\frac{8}{3}-\frac{5}{2}+1+\frac{1}{3}-\frac{1}{2}\right)\left(1+\frac{1}{2}\right)}}$$

Multiply \frac{125}{8} and \frac{4}{25} to get \frac{5}{2}.

$$\frac{\frac{\sqrt{1246}}{28}}{\sqrt{\frac{5}{2}-\left(\frac{8}{3}-\frac{5}{2}+1+\frac{1}{3}-\frac{1}{2}\right)\left(1+\frac{1}{2}\right)}}$$

Subtract \frac{5}{2} from \frac{8}{3} to get \frac{1}{6}.

$$\frac{\frac{\sqrt{1246}}{28}}{\sqrt{\frac{5}{2}-\left(\frac{1}{6}+1+\frac{1}{3}-\frac{1}{2}\right)\left(1+\frac{1}{2}\right)}}$$

Add \frac{1}{6} and 1 to get \frac{7}{6}.

$$\frac{\frac{\sqrt{1246}}{28}}{\sqrt{\frac{5}{2}-\left(\frac{7}{6}+\frac{1}{3}-\frac{1}{2}\right)\left(1+\frac{1}{2}\right)}}$$

Add \frac{7}{6} and \frac{1}{3} to get \frac{3}{2}.

$$\frac{\frac{\sqrt{1246}}{28}}{\sqrt{\frac{5}{2}-\left(\frac{3}{2}-\frac{1}{2}\right)\left(1+\frac{1}{2}\right)}}$$

Subtract \frac{1}{2} from \frac{3}{2} to get 1.

$$\frac{\frac{\sqrt{1246}}{28}}{\sqrt{\frac{5}{2}-1\left(1+\frac{1}{2}\right)}}$$

Add 1 and \frac{1}{2} to get \frac{3}{2}.

$$\frac{\frac{\sqrt{1246}}{28}}{\sqrt{\frac{5}{2}-1\times \frac{3}{2}}}$$

Multiply 1 and \frac{3}{2} to get \frac{3}{2}.

$$\frac{\frac{\sqrt{1246}}{28}}{\sqrt{\frac{5}{2}-\frac{3}{2}}}$$

Subtract \frac{3}{2} from \frac{5}{2} to get 1.

$$\frac{\frac{\sqrt{1246}}{28}}{\sqrt{1}}$$

Calculate the square root of 1 and get 1.

$$\frac{\frac{\sqrt{1246}}{28}}{1}$$

Anything divided by one gives itself.

$$\frac{\sqrt{1246}}{28}$$