Solve sqrt{(40+1/2)(2-1)}-(2/3-1)^-2:(1/5)^-1/(1-frac{1{4})^2}+frac{(frac{2-frac{1}{3}}{frac{3}{2}-1})^-2}{frac{frac{1}{2}-frac{2}{3}}{4-frac{2}{3}}}

$$\sqrt { ( 40 + \frac { 1 } { 2 } ) ( 2 - 1 ) } - \frac { ( \frac { 2 } { 3 } - 1 ) ^ { - 2 } : ( \frac { 1 } { 5 } ) ^ { - 1 } } { ( 1 - \frac { 1 } { 4 } ) ^ { 2 } } + \frac { ( \frac { 2 - \frac { 1 } { 3 } } { \frac { 3 } { 2 } - 1 } ) ^ { - 2 } } { \frac { \frac { 1 } { 2 } - \frac { 2 } { 3 } } { 4 - \frac { 2 } { 3 } } }$$

Evaluate

$$\frac{9\sqrt{2}}{2}-5\approx 1.363961031$$

Short Solution Steps

Add 40 and \frac{1}{2} to get \frac{81}{2}.

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

Subtract 1 from 2 to get 1.

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

Multiply \frac{81}{2} and 1 to get \frac{81}{2}.

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

Rewrite the square root of the division \sqrt{\frac{81}{2}} as the division of square roots \frac{\sqrt{81}}{\sqrt{2}}.

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

Calculate the square root of 81 and get 9.

$$\frac{9}{\sqrt{2}}-\frac{\frac{\left(\frac{2}{3}-1\right)^{-2}}{\left(\frac{1}{5}\right)^{-1}}}{\left(1-\frac{1}{4}\right)^{2}}+\frac{\left(\frac{2-\frac{1}{3}}{\frac{3}{2}-1}\right)^{-2}}{\frac{\frac{1}{2}-\frac{2}{3}}{4-\frac{2}{3}}}$$

Rationalize the denominator of \frac{9}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{9\sqrt{2}}{\left(\sqrt{2}\right)^{2}}-\frac{\frac{\left(\frac{2}{3}-1\right)^{-2}}{\left(\frac{1}{5}\right)^{-1}}}{\left(1-\frac{1}{4}\right)^{2}}+\frac{\left(\frac{2-\frac{1}{3}}{\frac{3}{2}-1}\right)^{-2}}{\frac{\frac{1}{2}-\frac{2}{3}}{4-\frac{2}{3}}}$$

The square of \sqrt{2} is 2.

$$\frac{9\sqrt{2}}{2}-\frac{\frac{\left(\frac{2}{3}-1\right)^{-2}}{\left(\frac{1}{5}\right)^{-1}}}{\left(1-\frac{1}{4}\right)^{2}}+\frac{\left(\frac{2-\frac{1}{3}}{\frac{3}{2}-1}\right)^{-2}}{\frac{\frac{1}{2}-\frac{2}{3}}{4-\frac{2}{3}}}$$

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

$$\frac{9\sqrt{2}}{2}-\frac{\frac{\left(-\frac{1}{3}\right)^{-2}}{\left(\frac{1}{5}\right)^{-1}}}{\left(1-\frac{1}{4}\right)^{2}}+\frac{\left(\frac{2-\frac{1}{3}}{\frac{3}{2}-1}\right)^{-2}}{\frac{\frac{1}{2}-\frac{2}{3}}{4-\frac{2}{3}}}$$

Calculate -\frac{1}{3} to the power of -2 and get 9.

$$\frac{9\sqrt{2}}{2}-\frac{\frac{9}{\left(\frac{1}{5}\right)^{-1}}}{\left(1-\frac{1}{4}\right)^{2}}+\frac{\left(\frac{2-\frac{1}{3}}{\frac{3}{2}-1}\right)^{-2}}{\frac{\frac{1}{2}-\frac{2}{3}}{4-\frac{2}{3}}}$$

Calculate \frac{1}{5} to the power of -1 and get 5.

$$\frac{9\sqrt{2}}{2}-\frac{\frac{9}{5}}{\left(1-\frac{1}{4}\right)^{2}}+\frac{\left(\frac{2-\frac{1}{3}}{\frac{3}{2}-1}\right)^{-2}}{\frac{\frac{1}{2}-\frac{2}{3}}{4-\frac{2}{3}}}$$

Subtract \frac{1}{4} from 1 to get \frac{3}{4}.

$$\frac{9\sqrt{2}}{2}-\frac{\frac{9}{5}}{\left(\frac{3}{4}\right)^{2}}+\frac{\left(\frac{2-\frac{1}{3}}{\frac{3}{2}-1}\right)^{-2}}{\frac{\frac{1}{2}-\frac{2}{3}}{4-\frac{2}{3}}}$$

Calculate \frac{3}{4} to the power of 2 and get \frac{9}{16}.

$$\frac{9\sqrt{2}}{2}-\frac{\frac{9}{5}}{\frac{9}{16}}+\frac{\left(\frac{2-\frac{1}{3}}{\frac{3}{2}-1}\right)^{-2}}{\frac{\frac{1}{2}-\frac{2}{3}}{4-\frac{2}{3}}}$$

Divide \frac{9}{5} by \frac{9}{16} by multiplying \frac{9}{5} by the reciprocal of \frac{9}{16}.

$$\frac{9\sqrt{2}}{2}-\frac{9}{5}\times \frac{16}{9}+\frac{\left(\frac{2-\frac{1}{3}}{\frac{3}{2}-1}\right)^{-2}}{\frac{\frac{1}{2}-\frac{2}{3}}{4-\frac{2}{3}}}$$

Multiply \frac{9}{5} and \frac{16}{9} to get \frac{16}{5}.

$$\frac{9\sqrt{2}}{2}-\frac{16}{5}+\frac{\left(\frac{2-\frac{1}{3}}{\frac{3}{2}-1}\right)^{-2}}{\frac{\frac{1}{2}-\frac{2}{3}}{4-\frac{2}{3}}}$$

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

$$\frac{9\sqrt{2}}{2}-\frac{16}{5}+\frac{\left(\frac{\frac{5}{3}}{\frac{3}{2}-1}\right)^{-2}}{\frac{\frac{1}{2}-\frac{2}{3}}{4-\frac{2}{3}}}$$

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

$$\frac{9\sqrt{2}}{2}-\frac{16}{5}+\frac{\left(\frac{\frac{5}{3}}{\frac{1}{2}}\right)^{-2}}{\frac{\frac{1}{2}-\frac{2}{3}}{4-\frac{2}{3}}}$$

Divide \frac{5}{3} by \frac{1}{2} by multiplying \frac{5}{3} by the reciprocal of \frac{1}{2}.

$$\frac{9\sqrt{2}}{2}-\frac{16}{5}+\frac{\left(\frac{5}{3}\times 2\right)^{-2}}{\frac{\frac{1}{2}-\frac{2}{3}}{4-\frac{2}{3}}}$$

Multiply \frac{5}{3} and 2 to get \frac{10}{3}.

$$\frac{9\sqrt{2}}{2}-\frac{16}{5}+\frac{\left(\frac{10}{3}\right)^{-2}}{\frac{\frac{1}{2}-\frac{2}{3}}{4-\frac{2}{3}}}$$

Calculate \frac{10}{3} to the power of -2 and get \frac{9}{100}.

$$\frac{9\sqrt{2}}{2}-\frac{16}{5}+\frac{\frac{9}{100}}{\frac{\frac{1}{2}-\frac{2}{3}}{4-\frac{2}{3}}}$$

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

$$\frac{9\sqrt{2}}{2}-\frac{16}{5}+\frac{\frac{9}{100}}{\frac{-\frac{1}{6}}{4-\frac{2}{3}}}$$

Subtract \frac{2}{3} from 4 to get \frac{10}{3}.

$$\frac{9\sqrt{2}}{2}-\frac{16}{5}+\frac{\frac{9}{100}}{\frac{-\frac{1}{6}}{\frac{10}{3}}}$$

Divide -\frac{1}{6} by \frac{10}{3} by multiplying -\frac{1}{6} by the reciprocal of \frac{10}{3}.

$$\frac{9\sqrt{2}}{2}-\frac{16}{5}+\frac{\frac{9}{100}}{-\frac{1}{6}\times \frac{3}{10}}$$

Multiply -\frac{1}{6} and \frac{3}{10} to get -\frac{1}{20}.

$$\frac{9\sqrt{2}}{2}-\frac{16}{5}+\frac{\frac{9}{100}}{-\frac{1}{20}}$$

Divide \frac{9}{100} by -\frac{1}{20} by multiplying \frac{9}{100} by the reciprocal of -\frac{1}{20}.

$$\frac{9\sqrt{2}}{2}-\frac{16}{5}+\frac{9}{100}\left(-20\right)$$

Multiply \frac{9}{100} and -20 to get -\frac{9}{5}.

$$\frac{9\sqrt{2}}{2}-\frac{16}{5}-\frac{9}{5}$$

Subtract \frac{9}{5} from -\frac{16}{5} to get -5.

$$\frac{9\sqrt{2}}{2}-5$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 5 times \frac{2}{2}.

$$\frac{9\sqrt{2}}{2}-\frac{5\times 2}{2}$$

Since \frac{9\sqrt{2}}{2} and \frac{5\times 2}{2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{9\sqrt{2}-5\times 2}{2}$$

Do the multiplications in 9\sqrt{2}-5\times 2.

$$\frac{9\sqrt{2}-10}{2}$$

Factor

$$\frac{9 \sqrt{2} - 10}{2} = 1.3639610306789285$$