(i) \( \frac{12^{4} \times 9^{3} \times 4}{6^{3} \times 8^{2} \times 27} \)

To simplify \( \frac{12^{4} \times 9^{3} \times 4}{6^{3} \times 8^{2} \times 27} \), let's first express everything in terms of prime factors: - \( 12 = 2^2 \times 3 \) so \( 12^4 = (2^2 \times 3)^4 = 2^8 \times 3^4 \) - \( 9 = 3^2 \) so \( 9^3 = (3^2)^3 = 3^6 \) - \( 4 = 2^2 \) Thus, the numerator becomes: \[ 12^{4} \times 9^{3} \times 4 = (2^8 \times 3^4) \times (3^6) \times (2^2) = 2^{10} \times 3^{10} \] Now for the denominator: - \( 6 = 2 \times 3 \) so \( 6^3 = (2 \times 3)^3 = 2^3 \times 3^3 \) - \( 8 = 2^3 \) so \( 8^2 = (2^3)^2 = 2^6 \) - \( 27 = 3^3 \) Thus, the denominator becomes: \[ 6^{3} \times 8^{2} \times 27 = (2^3 \times 3^3) \times (2^6) \times (3^3) = 2^{9} \times 3^{6} \] Now we can simplify the entire expression: \[ \frac{2^{10} \times 3^{10}}{2^{9} \times 3^{6}} = 2^{10-9} \times 3^{10-6} = 2^{1} \times 3^{4} = 2 \times 81 = 162 \] Thus, the simplified result is: \[ \frac{12^{4} \times 9^{3} \times 4}{6^{3} \times 8^{2} \times 27} = 162 \]