Solve {left({left({left(2-frac{3/9/3right)}^-2}{{left(9/4right)}^22^-1/5}right)}^right)}^-1

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

Evaluate

$$\frac{289}{160}=1.80625$$

Solution Steps

Express \frac{\frac{3}{9}}{3} as a single fraction.

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

Multiply 9 and 3 to get 27.

$$\left(\left(\frac{\left(2-\frac{3}{27}\right)^{-2}}{\left(\frac{9}{4}\right)^{2}\times \frac{2^{-1}}{5}}\right)^{1}\right)^{-1}$$

Reduce the fraction \frac{3}{27} to lowest terms by extracting and canceling out 3.

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

Subtract \frac{1}{9} from 2 to get \frac{17}{9}.

$$\left(\left(\frac{\left(\frac{17}{9}\right)^{-2}}{\left(\frac{9}{4}\right)^{2}\times \frac{2^{-1}}{5}}\right)^{1}\right)^{-1}$$

Calculate \frac{17}{9} to the power of -2 and get \frac{81}{289}.

$$\left(\left(\frac{\frac{81}{289}}{\left(\frac{9}{4}\right)^{2}\times \frac{2^{-1}}{5}}\right)^{1}\right)^{-1}$$

Calculate \frac{9}{4} to the power of 2 and get \frac{81}{16}.

$$\left(\left(\frac{\frac{81}{289}}{\frac{81}{16}\times \frac{2^{-1}}{5}}\right)^{1}\right)^{-1}$$

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

$$\left(\left(\frac{\frac{81}{289}}{\frac{81}{16}\times \frac{\frac{1}{2}}{5}}\right)^{1}\right)^{-1}$$

Express \frac{\frac{1}{2}}{5} as a single fraction.

$$\left(\left(\frac{\frac{81}{289}}{\frac{81}{16}\times \frac{1}{2\times 5}}\right)^{1}\right)^{-1}$$

Multiply 2 and 5 to get 10.

$$\left(\left(\frac{\frac{81}{289}}{\frac{81}{16}\times \frac{1}{10}}\right)^{1}\right)^{-1}$$

Multiply \frac{81}{16} and \frac{1}{10} to get \frac{81}{160}.

$$\left(\left(\frac{\frac{81}{289}}{\frac{81}{160}}\right)^{1}\right)^{-1}$$

Divide \frac{81}{289} by \frac{81}{160} by multiplying \frac{81}{289} by the reciprocal of \frac{81}{160}.

$$\left(\left(\frac{81}{289}\times \frac{160}{81}\right)^{1}\right)^{-1}$$

Multiply \frac{81}{289} and \frac{160}{81} to get \frac{160}{289}.

$$\left(\left(\frac{160}{289}\right)^{1}\right)^{-1}$$

Calculate \frac{160}{289} to the power of 1 and get \frac{160}{289}.

$$\left(\frac{160}{289}\right)^{-1}$$

Calculate \frac{160}{289} to the power of -1 and get \frac{289}{160}.

$$\frac{289}{160}$$

Factor

$$\frac{17 ^ {2}}{2 ^ {5} \cdot 5} = 1\frac{129}{160} = 1.80625$$