Solve ((frac{6^{3}}{6^{5}})^{2}*(6^2/6^4)^2)^4

$$( ( \frac { 6 ^ { 3 } } { 6 ^ { 5 } } ) ^ { 2 } * ( \frac { 6 ^ { 2 } } { 6 ^ { 4 } } ) ^ { 2 } ) ^ { 4 }$$

Evaluate

$$\frac{1}{7958661109946400884391936}\approx 1.256492752 \cdot 10^{-25}$$

Solution Steps

Rewrite 6^{5} as 6^{3}\times 6^{2}. Cancel out 6^{3} in both numerator and denominator.

$$\left(\left(\frac{1}{6^{2}}\right)^{2}\times \left(\frac{6^{2}}{6^{4}}\right)^{2}\right)^{4}$$

Rewrite 6^{4} as 6^{2}\times 6^{2}. Cancel out 6^{2} in both numerator and denominator.

$$\left(\left(\frac{1}{6^{2}}\right)^{2}\times \left(\frac{1}{6^{2}}\right)^{2}\right)^{4}$$

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

$$\left(\left(\frac{1}{6^{2}}\right)^{4}\right)^{4}$$

Calculate 6 to the power of 2 and get 36.

$$\left(\left(\frac{1}{36}\right)^{4}\right)^{4}$$

Calculate \frac{1}{36} to the power of 4 and get \frac{1}{1679616}.

$$\left(\frac{1}{1679616}\right)^{4}$$

Calculate \frac{1}{1679616} to the power of 4 and get \frac{1}{7958661109946400884391936}.

$$\frac{1}{7958661109946400884391936}$$

Factor

$$\frac{1}{2 ^ {32} \cdot 3 ^ {32}} = 1.2564927519658324 \times 10^{-25}$$