a) \( 6^{4} \times 3^{-4} \div 18 \)

To tackle the expression \( 6^{4} \times 3^{-4} \div 18 \), we can start by rewriting it for clarity. First, remember that \( 18 \) can be expressed as \( 6 \times 3 \). So the problem becomes \( 6^{4} \times 3^{-4} \div (6 \times 3) \). By using the properties of exponents, we know that \( 6^{4} \div 6 = 6^{3} \). So we have \( 6^{3} \times 3^{-4} \div 3 \), which simplifies to \( 6^{3} \times 3^{-5} \). This allows us to express the answer as a single fraction: \[ \frac{6^3}{3^5} \] Now calculating \( 6^3 = 216 \) and \( 3^5 = 243 \), the fraction simplifies to: \[ \frac{216}{243} \] Dividing both numerator and denominator by their greatest common divisor (which is 27), we arrive at: \[ \frac{8}{9} \] So, the final result is \( \frac{8}{9} \).