Solve frac{3^{-2}}{3^{-3}}*frac{2^{-2}}{2^{-2}}*frac{3^{4}}{2^{-5}}*2^4/3^5*2^-3/3^2

$$\frac { 3 ^ { - 2 } } { 3 ^ { - 3 } } \times \frac { 2 ^ { - 2 } } { 2 ^ { - 2 } } \times \frac { 3 ^ { 4 } } { 2 ^ { - 5 } } \times \frac { 2 ^ { 4 } } { 3 ^ { 5 } } \times \frac { 2 ^ { - 3 } } { 3 ^ { 2 } }$$

Evaluate

$$\frac{64}{9}\approx 7.111111111$$

Solution Steps

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract -3 from -2 to get 1.

$$3^{1}\times \frac{2^{-2}}{2^{-2}}\times \frac{3^{4}}{2^{-5}}\times \frac{2^{4}}{3^{5}}\times \frac{2^{-3}}{3^{2}}$$

Divide 2^{-2} by 2^{-2} to get 1.

$$3^{1}\times 1\times \frac{3^{4}}{2^{-5}}\times \frac{2^{4}}{3^{5}}\times \frac{2^{-3}}{3^{2}}$$

Calculate 3 to the power of 1 and get 3.

$$3\times 1\times \frac{3^{4}}{2^{-5}}\times \frac{2^{4}}{3^{5}}\times \frac{2^{-3}}{3^{2}}$$

Multiply 3 and 1 to get 3.

$$3\times \frac{3^{4}}{2^{-5}}\times \frac{2^{4}}{3^{5}}\times \frac{2^{-3}}{3^{2}}$$

Calculate 3 to the power of 4 and get 81.

$$3\times \frac{81}{2^{-5}}\times \frac{2^{4}}{3^{5}}\times \frac{2^{-3}}{3^{2}}$$

Calculate 2 to the power of -5 and get \frac{1}{32}.

$$3\times \frac{81}{\frac{1}{32}}\times \frac{2^{4}}{3^{5}}\times \frac{2^{-3}}{3^{2}}$$

Divide 81 by \frac{1}{32} by multiplying 81 by the reciprocal of \frac{1}{32}.

$$3\times 81\times 32\times \frac{2^{4}}{3^{5}}\times \frac{2^{-3}}{3^{2}}$$

Multiply 81 and 32 to get 2592.

$$3\times 2592\times \frac{2^{4}}{3^{5}}\times \frac{2^{-3}}{3^{2}}$$

Multiply 3 and 2592 to get 7776.

$$7776\times \frac{2^{4}}{3^{5}}\times \frac{2^{-3}}{3^{2}}$$

Calculate 2 to the power of 4 and get 16.

$$7776\times \frac{16}{3^{5}}\times \frac{2^{-3}}{3^{2}}$$

Calculate 3 to the power of 5 and get 243.

$$7776\times \frac{16}{243}\times \frac{2^{-3}}{3^{2}}$$

Multiply 7776 and \frac{16}{243} to get 512.

$$512\times \frac{2^{-3}}{3^{2}}$$

Calculate 2 to the power of -3 and get \frac{1}{8}.

$$512\times \frac{\frac{1}{8}}{3^{2}}$$

Calculate 3 to the power of 2 and get 9.

$$512\times \frac{\frac{1}{8}}{9}$$

Express \frac{\frac{1}{8}}{9} as a single fraction.

$$512\times \frac{1}{8\times 9}$$

Multiply 8 and 9 to get 72.

$$512\times \frac{1}{72}$$

Multiply 512 and \frac{1}{72} to get \frac{64}{9}.

$$\frac{64}{9}$$

Factor

$$\frac{2 ^ {6}}{3 ^ {2}} = 7\frac{1}{9} = 7.111111111111111$$