1. **State the problem:** Simplify the expression $$\frac{2^4 - (2^5)^8 - 2^6}{(2^4)^{33} - 16 - (2^7)^3 - 2^{10}}$$. 2. **Apply exponent rules:** Recall that $(a^m)^n = a^{m \times n}$ and $a^m \times a^n = a^{m+n}$. 3. **Simplify powers in numerator:** $$ (2^5)^8 = 2^{5 \times 8} = 2^{40} $$ So numerator becomes: $$ 2^4 - 2^{40} - 2^6 $$ 4. **Simplify powers in denominator:** $$ (2^4)^{33} = 2^{4 \times 33} = 2^{132} $$ $$ (2^7)^3 = 2^{7 \times 3} = 2^{21} $$ So denominator becomes: $$ 2^{132} - 16 - 2^{21} - 2^{10} $$ 5. **Rewrite 16 as power of 2:** $$ 16 = 2^4 $$ Denominator: $$ 2^{132} - 2^4 - 2^{21} - 2^{10} $$ 6. **Final simplified expression:** $$ \frac{2^4 - 2^{40} - 2^6}{2^{132} - 2^4 - 2^{21} - 2^{10}} $$ 7. **No common factors to cancel or further simplification possible due to large exponents and mixed terms.** **Answer:** $$ \boxed{\frac{2^4 - 2^{40} - 2^6}{2^{132} - 2^4 - 2^{21} - 2^{10}}} $$