Solve [frac{((2^{3})(2^{6}))^{-2}(3^{4})^{3}(3)}{((2^{6})(2^{10}))^{-1}(3^6)(3^2)(3^5)}]^10

$$[ \frac { ( ( 2 ^ { 3 } ) ( 2 ^ { 6 } ) ) ^ { - 2 } ( 3 ^ { 4 } ) ^ { 3 } ( 3 ) } { ( ( 2 ^ { 6 } ) ( 2 ^ { 10 } ) ) ^ { - 1 } ( 3 ^ { 6 } ) ( 3 ^ { 2 } ) ( 3 ^ { 5 } ) } ] ^ { 10 }$$

Evaluate

$$\frac{1}{1048576}\approx 0.000000954$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 3 and 6 to get 9.

$$\left(\frac{\left(2^{9}\right)^{-2}\times \left(3^{4}\right)^{3}\times 3}{\left(2^{6}\times 2^{10}\right)^{-1}\times 3^{6}\times 3^{2}\times 3^{5}}\right)^{10}$$

To raise a power to another power, multiply the exponents. Multiply 9 and -2 to get -18.

$$\left(\frac{2^{-18}\times \left(3^{4}\right)^{3}\times 3}{\left(2^{6}\times 2^{10}\right)^{-1}\times 3^{6}\times 3^{2}\times 3^{5}}\right)^{10}$$

To raise a power to another power, multiply the exponents. Multiply 4 and 3 to get 12.

$$\left(\frac{2^{-18}\times 3^{12}\times 3}{\left(2^{6}\times 2^{10}\right)^{-1}\times 3^{6}\times 3^{2}\times 3^{5}}\right)^{10}$$

To multiply powers of the same base, add their exponents. Add 12 and 1 to get 13.

$$\left(\frac{2^{-18}\times 3^{13}}{\left(2^{6}\times 2^{10}\right)^{-1}\times 3^{6}\times 3^{2}\times 3^{5}}\right)^{10}$$

To multiply powers of the same base, add their exponents. Add 6 and 10 to get 16.

$$\left(\frac{2^{-18}\times 3^{13}}{\left(2^{16}\right)^{-1}\times 3^{6}\times 3^{2}\times 3^{5}}\right)^{10}$$

To raise a power to another power, multiply the exponents. Multiply 16 and -1 to get -16.

$$\left(\frac{2^{-18}\times 3^{13}}{2^{-16}\times 3^{6}\times 3^{2}\times 3^{5}}\right)^{10}$$

To multiply powers of the same base, add their exponents. Add 6 and 2 to get 8.

$$\left(\frac{2^{-18}\times 3^{13}}{2^{-16}\times 3^{8}\times 3^{5}}\right)^{10}$$

To multiply powers of the same base, add their exponents. Add 8 and 5 to get 13.

$$\left(\frac{2^{-18}\times 3^{13}}{2^{-16}\times 3^{13}}\right)^{10}$$

Cancel out 3^{13} in both numerator and denominator.

$$\left(\frac{2^{-18}}{2^{-16}}\right)^{10}$$

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

$$\left(\frac{1}{2^{2}}\right)^{10}$$

Calculate 2 to the power of 2 and get 4.

$$\left(\frac{1}{4}\right)^{10}$$

Calculate \frac{1}{4} to the power of 10 and get \frac{1}{1048576}.

$$\frac{1}{1048576}$$

Factor

$$\frac{1}{2 ^ {20}} = 9.5367431640625 \times 10^{-7}$$