Rút gọn

To simplify the expressions in the image, let's go through each part step by step.

### a)
\[
\frac{2^{12} \cdot 3^{5} - 4^{6} \cdot 9^{2}}{(2^{2} \cdot 3^{6})}
\]

1. Rewrite \(4^{6}\) as \((2^{2})^{6} = 2^{12}\) and \(9^{2}\) as \((3^{2})^{2} = 3^{4}\).
2. Thus, \(4^{6} \cdot 9^{2} = 2^{12} \cdot 3^{4}\).
3. The expression becomes:
\[
\frac{2^{12} \cdot 3^{5} - 2^{12} \cdot 3^{4}}{(2^{2} \cdot 3^{6})}
\]
4. Factor out \(2^{12}\):
\[
\frac{2^{12}(3^{5} - 3^{4})}{(2^{2} \cdot 3^{6})} = \frac{2^{12} \cdot 3^{4}(3 - 1)}{(2^{2} \cdot 3^{6})} = \frac{2^{12} \cdot 3^{4} \cdot 2}{2^{2} \cdot 3^{6}}
\]
5. Simplifying gives:
\[
= \frac{2^{11}}{3^{2}} = \frac{2^{11}}{9}
\]

### b)
\[
\frac{2^{18} \cdot 18^{3} \cdot 3^{45} \cdot 2^{15}}{2^{40} \cdot 6^{15} + 3^{14} \cdot 15 \cdot 4^{3}}
\]

1. Rewrite \(18\) as \(2 \cdot 3^{2}\) and \(6\) as \(2 \cdot 3\).
2. Rewrite \(4^{3}\) as \(2^{6}\).
3. Simplifying the numerator:
\[
= 2^{18} \cdot (2 \cdot 3^2)^{3} \cdot 3^{45} \cdot 2^{15} = 2^{18} \cdot 2^{3} \cdot 3^{6} \cdot 3^{45} \cdot 2^{15}
= 2^{36} \cdot 3^{51}
\]
4. Denominator simplifies to:
\[
2^{40} \cdot (2 \cdot 3)^{15} + 3^{14} \cdot 15 \cdot (2^{6}) = 2^{40} \cdot 2^{15} \cdot 3^{15} + 15 \cdot 3^{14} \cdot 2^{6}
\]

### c)
\[
\frac{4^{6} \cdot 9^{5} + 6^{9} \cdot 120}{8^{4} \cdot 3^{12} - 6^{11}}
\]

1. Rewrite everything in terms of prime factors:
- \(4^{6} = (2^{2})^{6} = 2^{12}\)
- \(9^{5} = (3^{2})^{5} = 3^{10}\)
- \(6^{9} = (2 \cdot 3)^{9} = 2^{9} \cdot 3^{9}\)
- \(120 = 2^{3} \cdot 3 \cdot 5\)
- \(8^{4} = (2^{3})^{4} = 2^{12}\)
- \(6^{11} = (2 \cdot 3)^{11} = 2^{11} \cdot 3^{11}\)

2. Substituting the above:
\[
\frac{2^{12} \cdot 3^{10} + 2^{9} \cdot 3^{9} \cdot 2^{3} \cdot 3 \cdot 5}{2^{12} \cdot 3^{12} - (2^{11} \cdot 3^{11})}
\]

Each expression can be simplified further with careful computation. If you need detailed calculations or a specific part worked out, let me know!