(9^-3 cdot 27^6)/(81^4)

To simplify the expression \( \frac{9^{-3} \cdot 27^{6}}{81^{4}} \), we can rewrite each term using powers of 3:<br /><br />\( 9 = 3^2 \), \( 27 = 3^3 \), and \( 81 = 3^4 \).<br /><br />Substituting these values, we get:<br /><br />\( \frac{(3^2)^{-3} \cdot (3^3)^6}{(3^4)^4} \)<br /><br />Simplifying the exponents, we have:<br /><br />\( \frac{3^{-6} \cdot 3^{18}}{3^{16}} \)<br /><br />Using the properties of exponents, we can combine the terms in the numerator:<br /><br />\( \frac{3^{-6 + 18}}{3^{16}} \)<br /><br />Simplifying the exponent in the numerator, we get:<br /><br />\( \frac{3^{12}}{3^{16}} \)<br /><br />Using the properties of exponents again, we can simplify the fraction:<br /><br />\( 3^{12 - 16} = 3^{-4} \)<br /><br />Finally, we can rewrite \( 3^{-4} \) as \( \frac{1}{3^4} \), which is equal to \( \frac{1}{81} \).<br /><br />Therefore, the simplified form of the expression \( \frac{9^{-3} \cdot 27^{6}}{81^{4}} \) is \( \frac{1}{81} \).