Solve (frac{2/3)^{5}(2/3)^{0}(2/3)^{-3}(81/16)^{-2}}{(frac{3}{2})^-5(frac{2}{3})[(frac{2}{3})^5]^2(frac{8}{27})^3}=

$$\frac { ( \frac { 2 } { 3 } ) ^ { 5 } ( \frac { 2 } { 3 } ) ^ { 0 } ( \frac { 2 } { 3 } ) ^ { - 3 } ( \frac { 81 } { 16 } ) ^ { - 2 } } { ( \frac { 3 } { 2 } ) ^ { - 5 } ( \frac { 2 } { 3 } ) [ ( \frac { 2 } { 3 } ) ^ { 5 } ] ^ { 2 } ( \frac { 8 } { 27 } ) ^ { 3 } } =$$

Evaluate

$$\frac{14348907}{32768}=437.893890381$$

Solution Steps

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

$$\frac{\left(\frac{2}{3}\right)^{5}\times \left(\frac{2}{3}\right)^{-3}\times \left(\frac{81}{16}\right)^{-2}}{\left(\frac{3}{2}\right)^{-5}\times \frac{2}{3}\times \left(\left(\frac{2}{3}\right)^{5}\right)^{2}\times \left(\frac{8}{27}\right)^{3}}$$

To multiply powers of the same base, add their exponents. Add 5 and -3 to get 2.

$$\frac{\left(\frac{2}{3}\right)^{2}\times \left(\frac{81}{16}\right)^{-2}}{\left(\frac{3}{2}\right)^{-5}\times \frac{2}{3}\times \left(\left(\frac{2}{3}\right)^{5}\right)^{2}\times \left(\frac{8}{27}\right)^{3}}$$

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

$$\frac{\left(\frac{2}{3}\right)^{2}\times \left(\frac{81}{16}\right)^{-2}}{\left(\frac{3}{2}\right)^{-5}\times \frac{2}{3}\times \left(\frac{2}{3}\right)^{10}\times \left(\frac{8}{27}\right)^{3}}$$

To multiply powers of the same base, add their exponents. Add 1 and 10 to get 11.

$$\frac{\left(\frac{2}{3}\right)^{2}\times \left(\frac{81}{16}\right)^{-2}}{\left(\frac{3}{2}\right)^{-5}\times \left(\frac{2}{3}\right)^{11}\times \left(\frac{8}{27}\right)^{3}}$$

Cancel out \left(\frac{2}{3}\right)^{2} in both numerator and denominator.

$$\frac{\left(\frac{81}{16}\right)^{-2}}{\left(\frac{8}{27}\right)^{3}\times \left(\frac{2}{3}\right)^{9}\times \left(\frac{3}{2}\right)^{-5}}$$

Calculate \frac{81}{16} to the power of -2 and get \frac{256}{6561}.

$$\frac{\frac{256}{6561}}{\left(\frac{8}{27}\right)^{3}\times \left(\frac{2}{3}\right)^{9}\times \left(\frac{3}{2}\right)^{-5}}$$

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

$$\frac{\frac{256}{6561}}{\frac{512}{19683}\times \left(\frac{2}{3}\right)^{9}\times \left(\frac{3}{2}\right)^{-5}}$$

Calculate \frac{2}{3} to the power of 9 and get \frac{512}{19683}.

$$\frac{\frac{256}{6561}}{\frac{512}{19683}\times \frac{512}{19683}\times \left(\frac{3}{2}\right)^{-5}}$$

Multiply \frac{512}{19683} and \frac{512}{19683} to get \frac{262144}{387420489}.

$$\frac{\frac{256}{6561}}{\frac{262144}{387420489}\times \left(\frac{3}{2}\right)^{-5}}$$

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

$$\frac{\frac{256}{6561}}{\frac{262144}{387420489}\times \frac{32}{243}}$$

Multiply \frac{262144}{387420489} and \frac{32}{243} to get \frac{8388608}{94143178827}.

$$\frac{\frac{256}{6561}}{\frac{8388608}{94143178827}}$$

Divide \frac{256}{6561} by \frac{8388608}{94143178827} by multiplying \frac{256}{6561} by the reciprocal of \frac{8388608}{94143178827}.

$$\frac{256}{6561}\times \frac{94143178827}{8388608}$$

Multiply \frac{256}{6561} and \frac{94143178827}{8388608} to get \frac{14348907}{32768}.

$$\frac{14348907}{32768}$$

Factor

$$\frac{3 ^ {15}}{2 ^ {15}} = 437\frac{29291}{32768} = 437.8938903808594$$