1. **State the problem:** Simplify the expression $$\frac{2^{3}-\left(2^{5}\right)^{8}-2^{6}}{\left(2^{4}\right)^{3}-16-\left(2^{7}\right)^{3}-2^{10}}$$. 2. **Recall exponent rules:** - Power of a power: $$\left(a^{m}\right)^{n} = a^{m \times n}$$ - Subtraction and addition are performed after simplification of powers. 3. **Simplify each term:** - $$2^{3} = 8$$ - $$\left(2^{5}\right)^{8} = 2^{5 \times 8} = 2^{40}$$ - $$2^{6} = 64$$ - $$\left(2^{4}\right)^{3} = 2^{4 \times 3} = 2^{12}$$ - $$16 = 2^{4}$$ (for comparison) - $$\left(2^{7}\right)^{3} = 2^{7 \times 3} = 2^{21}$$ - $$2^{10} = 1024$$ 4. **Rewrite numerator and denominator:** $$\text{Numerator} = 8 - 2^{40} - 64$$ $$\text{Denominator} = 2^{12} - 16 - 2^{21} - 2^{10}$$ 5. **Rewrite constants as powers of 2:** - $$8 = 2^{3}$$ - $$64 = 2^{6}$$ - $$16 = 2^{4}$$ - $$1024 = 2^{10}$$ 6. **Numerator:** $$2^{3} - 2^{40} - 2^{6}$$ 7. **Denominator:** $$2^{12} - 2^{4} - 2^{21} - 2^{10}$$ 8. **Factor out the smallest power in numerator and denominator if possible:** - Numerator smallest power is $$2^{3}$$ - Denominator smallest power is $$2^{4}$$ 9. **Factor numerator:** $$2^{3} - 2^{6} - 2^{40} = 2^{3} \left(1 - 2^{3} - 2^{37}\right)$$ 10. **Factor denominator:** $$2^{4} - 2^{10} - 2^{12} - 2^{21} = 2^{4} \left(1 - 2^{6} - 2^{8} - 2^{17}\right)$$ 11. **Rewrite the fraction:** $$\frac{2^{3} \left(1 - 2^{3} - 2^{37}\right)}{2^{4} \left(1 - 2^{6} - 2^{8} - 2^{17}\right)}$$ 12. **Cancel common factor $$2^{3}$$:** $$= \frac{\cancel{2^{3}} \left(1 - 2^{3} - 2^{37}\right)}{2^{4} \left(1 - 2^{6} - 2^{8} - 2^{17}\right)} = \frac{1 - 2^{3} - 2^{37}}{2^{1} \left(1 - 2^{6} - 2^{8} - 2^{17}\right)}$$ 13. **Final simplified form:** $$\boxed{\frac{1 - 2^{3} - 2^{37}}{2 \left(1 - 2^{6} - 2^{8} - 2^{17}\right)}}$$ This is the simplest exact form without approximating large powers.