Solve frac{6^{2}*12^{2}*27^{-2}*16^{-3}}{4^{-1}*3^{5}*12^{-1}*6^{3}}*frac{3^{5}*12^{-4}*8^3}{4^3*24^-1*8^2}=

$$\frac { 6 ^ { 2 } \cdot 12 ^ { 2 } \cdot 27 ^ { - 2 } \cdot 16 ^ { - 3 } } { 4 ^ { - 1 } \cdot 3 ^ { 5 } \cdot 12 ^ { - 1 } \cdot 6 ^ { 3 } } \cdot \frac { 3 ^ { 5 } \cdot 12 ^ { - 4 } \cdot 8 ^ { 3 } } { 4 ^ { 3 } \cdot 24 ^ { - 1 } \cdot 8 ^ { 2 } } =$$

Evaluate

$$\frac{1}{17915904}\approx 0.000000056$$

Solution Steps

Cancel out 6^{2} in both numerator and denominator.

$$\frac{12^{2}\times 16^{-3}\times 27^{-2}}{6\times 3^{5}\times \frac{1}{4}\times \frac{1}{12}}\times \frac{3^{5}\times 12^{-4}\times 8^{3}}{4^{3}\times 24^{-1}\times 8^{2}}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$\frac{12^{3}\times 16^{-3}\times 27^{-2}}{6\times 3^{5}\times \frac{1}{4}}\times \frac{3^{5}\times 12^{-4}\times 8^{3}}{4^{3}\times 24^{-1}\times 8^{2}}$$

Calculate 12 to the power of 3 and get 1728.

$$\frac{1728\times 16^{-3}\times 27^{-2}}{6\times 3^{5}\times \frac{1}{4}}\times \frac{3^{5}\times 12^{-4}\times 8^{3}}{4^{3}\times 24^{-1}\times 8^{2}}$$

Calculate 16 to the power of -3 and get \frac{1}{4096}.

$$\frac{1728\times \frac{1}{4096}\times 27^{-2}}{6\times 3^{5}\times \frac{1}{4}}\times \frac{3^{5}\times 12^{-4}\times 8^{3}}{4^{3}\times 24^{-1}\times 8^{2}}$$

Multiply 1728 and \frac{1}{4096} to get \frac{27}{64}.

$$\frac{\frac{27}{64}\times 27^{-2}}{6\times 3^{5}\times \frac{1}{4}}\times \frac{3^{5}\times 12^{-4}\times 8^{3}}{4^{3}\times 24^{-1}\times 8^{2}}$$

Calculate 27 to the power of -2 and get \frac{1}{729}.

$$\frac{\frac{27}{64}\times \frac{1}{729}}{6\times 3^{5}\times \frac{1}{4}}\times \frac{3^{5}\times 12^{-4}\times 8^{3}}{4^{3}\times 24^{-1}\times 8^{2}}$$

Multiply \frac{27}{64} and \frac{1}{729} to get \frac{1}{1728}.

$$\frac{\frac{1}{1728}}{6\times 3^{5}\times \frac{1}{4}}\times \frac{3^{5}\times 12^{-4}\times 8^{3}}{4^{3}\times 24^{-1}\times 8^{2}}$$

Calculate 3 to the power of 5 and get 243.

$$\frac{\frac{1}{1728}}{6\times 243\times \frac{1}{4}}\times \frac{3^{5}\times 12^{-4}\times 8^{3}}{4^{3}\times 24^{-1}\times 8^{2}}$$

Multiply 6 and 243 to get 1458.

$$\frac{\frac{1}{1728}}{1458\times \frac{1}{4}}\times \frac{3^{5}\times 12^{-4}\times 8^{3}}{4^{3}\times 24^{-1}\times 8^{2}}$$

Multiply 1458 and \frac{1}{4} to get \frac{729}{2}.

$$\frac{\frac{1}{1728}}{\frac{729}{2}}\times \frac{3^{5}\times 12^{-4}\times 8^{3}}{4^{3}\times 24^{-1}\times 8^{2}}$$

Divide \frac{1}{1728} by \frac{729}{2} by multiplying \frac{1}{1728} by the reciprocal of \frac{729}{2}.

$$\frac{1}{1728}\times \frac{2}{729}\times \frac{3^{5}\times 12^{-4}\times 8^{3}}{4^{3}\times 24^{-1}\times 8^{2}}$$

Multiply \frac{1}{1728} and \frac{2}{729} to get \frac{1}{629856}.

$$\frac{1}{629856}\times \frac{3^{5}\times 12^{-4}\times 8^{3}}{4^{3}\times 24^{-1}\times 8^{2}}$$

Cancel out 8^{2} in both numerator and denominator.

$$\frac{1}{629856}\times \frac{8\times 3^{5}\times 12^{-4}}{4^{3}\times \frac{1}{24}}$$

Calculate 3 to the power of 5 and get 243.

$$\frac{1}{629856}\times \frac{8\times 243\times 12^{-4}}{4^{3}\times \frac{1}{24}}$$

Multiply 8 and 243 to get 1944.

$$\frac{1}{629856}\times \frac{1944\times 12^{-4}}{4^{3}\times \frac{1}{24}}$$

Calculate 12 to the power of -4 and get \frac{1}{20736}.

$$\frac{1}{629856}\times \frac{1944\times \frac{1}{20736}}{4^{3}\times \frac{1}{24}}$$

Multiply 1944 and \frac{1}{20736} to get \frac{3}{32}.

$$\frac{1}{629856}\times \frac{\frac{3}{32}}{4^{3}\times \frac{1}{24}}$$

Calculate 4 to the power of 3 and get 64.

$$\frac{1}{629856}\times \frac{\frac{3}{32}}{64\times \frac{1}{24}}$$

Multiply 64 and \frac{1}{24} to get \frac{8}{3}.

$$\frac{1}{629856}\times \frac{\frac{3}{32}}{\frac{8}{3}}$$

Divide \frac{3}{32} by \frac{8}{3} by multiplying \frac{3}{32} by the reciprocal of \frac{8}{3}.

$$\frac{1}{629856}\times \frac{3}{32}\times \frac{3}{8}$$

Multiply \frac{3}{32} and \frac{3}{8} to get \frac{9}{256}.

$$\frac{1}{629856}\times \frac{9}{256}$$

Multiply \frac{1}{629856} and \frac{9}{256} to get \frac{1}{17915904}.

$$\frac{1}{17915904}$$

Factor

$$\frac{1}{2 ^ {13} \cdot 3 ^ {7}} = 5.581632944673068 \times 10^{-8}$$