Sure, let's simplify each expression step by step.

1) \(\frac{3^{50} \cdot 5^{44} \cdot 11^{28}}{5^{43} \cdot 3^{54} \cdot 11^{30}}\)

- Simplify each base:
- \(3^{50} \div 3^{54} = 3^{-4}\)
- \(5^{44} \div 5^{43} = 5^{1}\)
- \(11^{28} \div 11^{30} = 11^{-2}\)

- Combined result: \(3^{-4} \cdot 5 \cdot 11^{-2}\)

2) \(2^3 \cdot 5^3 \cdot 3 \cdot \left[539 - \left(639 - 8 \cdot (7^8 \cdot 7^6 \cdot 2021^0)\right)\right]\)

- Note that \(2021^0 = 1\).
- Simplify the inner expression: \(7^8 \cdot 7^6 = 7^{14}\).
- So, the expression becomes \(2^3 \cdot 5^3 \cdot 3 \cdot \left[539 - \left(639 - 8 \cdot 7^{14}\right)\right]\).

3) \(\frac{3^{11} \cdot 11 + 3^{11} \cdot 21}{3^9 \cdot 2^8}\)

- Factor out \(3^{11}\) from the numerator: \(3^{11} \cdot (11 + 21) = 3^{11} \cdot 32\).
- Divide: \(\frac{3^{11} \cdot 32}{3^9 \cdot 2^8} = 3^{2} \cdot \frac{32}{2^8} = 3^2 \cdot \frac{32}{256} = 3^2 \cdot \frac{1}{8}\).

4) \(\frac{5 \cdot 4^{15} \cdot 9^9 - 4 \cdot 3^{20} \cdot 8^9}{5 \cdot 2^9 \cdot 6^{19} - 7 \cdot 2^{29} \cdot 27^6}\)

- Simplify numbers in base:
- \(4^{15} = (2^2)^{15} = 2^{30}\).
- \(9^9 = (3^2)^9 = 3^{18}\).
- \(8^9 = (2^3)^9 = 2^{27}\).
- \(6^{19} = (2 \cdot 3)^{19} = 2^{19} \cdot 3^{19}\).
- \(27^6 = (3^3)^6 = 3^{18}\).

- Now substitute these values:
\[\frac{5 \cdot 2^{30} \cdot 3^{18} - 4 \cdot 3^{20} \cdot 2^{27}}{5 \cdot 2^9 \cdot 2^{19} \cdot 3^{19} - 7 \cdot 2^{29} \cdot 3^{18}}\]

- Simplify to \(\frac{5 \cdot 2^{30} \cdot 3^{18} - 4 \cdot 2^{27} \cdot 3^{20}}{5 \cdot 2^{28} \cdot 3^{19} - 7 \cdot 2^{29} \cdot 3^{18}}\).

5) \(\frac{2^7 \cdot 3 + 2^{10}}{13 \cdot 2^7 - 14 \cdot 2^6}\)

- Factor the numerator: \(2^7 (3 + 2^3)\).
- Simplify the denominator: \(2^6 (13 \cdot 2 - 14)\).

- This simplifies to \(\frac{2^7 \cdot 11}{2^6 \cdot 12} = \frac{11}{2}.\)

These steps should help simplify each expression!