(9^-3 cdot 27^6)/(84^4)=

To simplify the expression \( \frac{9^{-3} \cdot 27^{6}}{84^{4}} \), we can start by expressing all the numbers in terms of their prime factors.<br /><br />First, let's rewrite \( 9 \) and \( 27 \) as powers of \( 3 \):<br />\[ 9 = 3^2 \]<br />\[ 27 = 3^3 \]<br /><br />Now, substitute these into the original expression:<br />\[ \frac{(3^2)^{-3} \cdot (3^3)^6}{84^4} \]<br /><br />Next, simplify the exponents:<br />\[ (3^2)^{-3} = 3^{2 \cdot (-3)} = 3^{-6} \]<br />\[ (3^3)^6 = 3^{3 \cdot 6} = 3^{18} \]<br /><br />So the expression becomes:<br />\[ \frac{3^{-6} \cdot 3^{18}}{84^4} \]<br /><br />Combine the exponents in the numerator:<br />\[ 3^{-6} \cdot 3^{18} = 3^{-6 + 18} = 3^{12} \]<br /><br />Now the expression is:<br />\[ \frac{3^{12}}{84^4} \]<br /><br />Next, express \( 84 \) in terms of its prime factors:<br />\[ 84 = 2^2 \cdot 3 \cdot 7 \]<br /><br />So,<br />\[ 84^4 = (2^2 \cdot 3 \cdot 7)^4 = 2^{8} \cdot 3^4 \cdot 7^4 \]<br /><br />Now the expression becomes:<br />\[ \frac{3^{12}}{2^8 \cdot 3^4 \cdot 7^4} \]<br /><br />Simplify the fraction by canceling out the common factors in the numerator and the denominator:<br />\[ \frac{3^{12}}{2^8 \cdot 3^4 \cdot 7^4} = \frac{3^{12-4}}{2^8 \cdot 7^4} = \frac{3^8}{2^8 \cdot 7^4} \]<br /><br />Finally, simplify the expression:<br />\[ \frac{3^8}{2^8 \cdot 7^4} = \left( \frac{3}{2 \cdot 7} \right)^8 = \left( \frac{3}{14} \right)^8 \]<br /><br />Thus, the simplified form of the expression is:<br />\[ \left( \frac{3}{14} \right)^8 \]