Solve (frac{2/3)^{5}(2/3)^{0}(2/3)^{3}(81/6)^{-2}}{(frac{3}{2})^{-5}(frac{2}{3})^1}[(frac{2}{3})^5]^2(frac{8}{27})^3

$$\frac { ( \frac { 2 } { 3 } ) ^ { 5 } ( \frac { 2 } { 3 } ) ^ { 0 } ( \frac { 2 } { 3 } ) ^ { 3 } ( \frac { 81 } { 6 } ) ^ { - 2 } } { ( \frac { 3 } { 2 } ) ^ { - 5 } ( \frac { 2 } { 3 } ) ^ { 1 } } [ ( \frac { 2 } { 3 } ) ^ { 5 } ] ^ { 2 } ( \frac { 8 } { 27 } ) ^ { 3 }$$

Evaluate

$$\frac{8388608}{7625597484987}\approx 0.0000011$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 5 and 0 to get 5.

$$\frac{\left(\frac{2}{3}\right)^{5}\times \left(\frac{2}{3}\right)^{3}\times \left(\frac{81}{6}\right)^{-2}}{\left(\frac{3}{2}\right)^{-5}\times \left(\frac{2}{3}\right)^{1}}\times \left(\left(\frac{2}{3}\right)^{5}\right)^{2}\times \left(\frac{8}{27}\right)^{3}$$

To multiply powers of the same base, add their exponents. Add 5 and 3 to get 8.

$$\frac{\left(\frac{2}{3}\right)^{8}\times \left(\frac{81}{6}\right)^{-2}}{\left(\frac{3}{2}\right)^{-5}\times \left(\frac{2}{3}\right)^{1}}\times \left(\left(\frac{2}{3}\right)^{5}\right)^{2}\times \left(\frac{8}{27}\right)^{3}$$

To raise a power to another power, multiply the exponents. Multiply 5 and 2 to get 10.

$$\frac{\left(\frac{2}{3}\right)^{8}\times \left(\frac{81}{6}\right)^{-2}}{\left(\frac{3}{2}\right)^{-5}\times \left(\frac{2}{3}\right)^{1}}\times \left(\frac{2}{3}\right)^{10}\times \left(\frac{8}{27}\right)^{3}$$

Calculate \frac{2}{3} to the power of 8 and get \frac{256}{6561}.

$$\frac{\frac{256}{6561}\times \left(\frac{81}{6}\right)^{-2}}{\left(\frac{3}{2}\right)^{-5}\times \left(\frac{2}{3}\right)^{1}}\times \left(\frac{2}{3}\right)^{10}\times \left(\frac{8}{27}\right)^{3}$$

Reduce the fraction \frac{81}{6} to lowest terms by extracting and canceling out 3.

$$\frac{\frac{256}{6561}\times \left(\frac{27}{2}\right)^{-2}}{\left(\frac{3}{2}\right)^{-5}\times \left(\frac{2}{3}\right)^{1}}\times \left(\frac{2}{3}\right)^{10}\times \left(\frac{8}{27}\right)^{3}$$

Calculate \frac{27}{2} to the power of -2 and get \frac{4}{729}.

$$\frac{\frac{256}{6561}\times \frac{4}{729}}{\left(\frac{3}{2}\right)^{-5}\times \left(\frac{2}{3}\right)^{1}}\times \left(\frac{2}{3}\right)^{10}\times \left(\frac{8}{27}\right)^{3}$$

Multiply \frac{256}{6561} and \frac{4}{729} to get \frac{1024}{4782969}.

$$\frac{\frac{1024}{4782969}}{\left(\frac{3}{2}\right)^{-5}\times \left(\frac{2}{3}\right)^{1}}\times \left(\frac{2}{3}\right)^{10}\times \left(\frac{8}{27}\right)^{3}$$

Calculate \frac{3}{2} to the power of -5 and get \frac{32}{243}.

$$\frac{\frac{1024}{4782969}}{\frac{32}{243}\times \left(\frac{2}{3}\right)^{1}}\times \left(\frac{2}{3}\right)^{10}\times \left(\frac{8}{27}\right)^{3}$$

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

$$\frac{\frac{1024}{4782969}}{\frac{32}{243}\times \frac{2}{3}}\times \left(\frac{2}{3}\right)^{10}\times \left(\frac{8}{27}\right)^{3}$$

Multiply \frac{32}{243} and \frac{2}{3} to get \frac{64}{729}.

$$\frac{\frac{1024}{4782969}}{\frac{64}{729}}\times \left(\frac{2}{3}\right)^{10}\times \left(\frac{8}{27}\right)^{3}$$

Divide \frac{1024}{4782969} by \frac{64}{729} by multiplying \frac{1024}{4782969} by the reciprocal of \frac{64}{729}.

$$\frac{1024}{4782969}\times \frac{729}{64}\times \left(\frac{2}{3}\right)^{10}\times \left(\frac{8}{27}\right)^{3}$$

Multiply \frac{1024}{4782969} and \frac{729}{64} to get \frac{16}{6561}.

$$\frac{16}{6561}\times \left(\frac{2}{3}\right)^{10}\times \left(\frac{8}{27}\right)^{3}$$

Calculate \frac{2}{3} to the power of 10 and get \frac{1024}{59049}.

$$\frac{16}{6561}\times \frac{1024}{59049}\times \left(\frac{8}{27}\right)^{3}$$

Multiply \frac{16}{6561} and \frac{1024}{59049} to get \frac{16384}{387420489}.

$$\frac{16384}{387420489}\times \left(\frac{8}{27}\right)^{3}$$

Calculate \frac{8}{27} to the power of 3 and get \frac{512}{19683}.

$$\frac{16384}{387420489}\times \frac{512}{19683}$$

Multiply \frac{16384}{387420489} and \frac{512}{19683} to get \frac{8388608}{7625597484987}.

$$\frac{8388608}{7625597484987}$$

Factor

$$\frac{2 ^ {23}}{3 ^ {27}} = 1.100059112287947 \times 10^{-6}$$