Solve frac{3^{2}*9^{-1}*(3^{3}*9^{-2})^{2}*(3^{-1}*frac{1^{-3}}{3})^{-2}}{(3^2*9^2*2^-4)^-1}

$$\frac { 3 ^ { 2 } \cdot 9 ^ { - 1 } \cdot ( 3 ^ { 3 } \cdot 9 ^ { - 2 } ) ^ { 2 } \cdot ( 3 ^ { - 1 } \cdot \frac { 1 ^ { - 3 } } { 3 } ) ^ { - 2 } } { ( 3 ^ { 2 } \cdot 9 ^ { 2 } \cdot 2 ^ { - 4 } ) ^ { - 1 } }$$

Evaluate

$$\frac{6561}{16}=410.0625$$

Solution Steps

Calculate 3 to the power of 2 and get 9.

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

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

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

Multiply 9 and \frac{1}{9} to get 1.

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

Calculate 3 to the power of 3 and get 27.

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

Calculate 9 to the power of -2 and get \frac{1}{81}.

$$\frac{\left(27\times \frac{1}{81}\right)^{2}\times \left(3^{-1}\times \frac{1^{-3}}{3}\right)^{-2}}{\left(3^{2}\times 9^{2}\times 2^{-4}\right)^{-1}}$$

Multiply 27 and \frac{1}{81} to get \frac{1}{3}.

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

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

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

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

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

Calculate 1 to the power of -3 and get 1.

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

Multiply \frac{1}{3} and \frac{1}{3} to get \frac{1}{9}.

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

Calculate \frac{1}{9} to the power of -2 and get 81.

$$\frac{\frac{1}{9}\times 81}{\left(3^{2}\times 9^{2}\times 2^{-4}\right)^{-1}}$$

Multiply \frac{1}{9} and 81 to get 9.

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

Calculate 3 to the power of 2 and get 9.

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

Calculate 9 to the power of 2 and get 81.

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

Multiply 9 and 81 to get 729.

$$\frac{9}{\left(729\times 2^{-4}\right)^{-1}}$$

Calculate 2 to the power of -4 and get \frac{1}{16}.

$$\frac{9}{\left(729\times \frac{1}{16}\right)^{-1}}$$

Multiply 729 and \frac{1}{16} to get \frac{729}{16}.

$$\frac{9}{\left(\frac{729}{16}\right)^{-1}}$$

Calculate \frac{729}{16} to the power of -1 and get \frac{16}{729}.

$$\frac{9}{\frac{16}{729}}$$

Divide 9 by \frac{16}{729} by multiplying 9 by the reciprocal of \frac{16}{729}.

$$9\times \frac{729}{16}$$

Multiply 9 and \frac{729}{16} to get \frac{6561}{16}.

$$\frac{6561}{16}$$

Factor

$$\frac{3 ^ {8}}{2 ^ {4}} = 410\frac{1}{16} = 410.0625$$