1. **State the problem:** We need to determine which number is larger: $$0.2^{2^{15}}$$ or $$0.05^{2^{10}}$$. 2. **Recall the properties of exponents:** For positive numbers less than 1, raising them to larger powers results in smaller numbers. 3. **Calculate the exponents:** - $$2^{15} = 32768$$ - $$2^{10} = 1024$$ 4. **Rewrite the bases as powers of 10:** - $$0.2 = 2 \times 10^{-1}$$ - $$0.05 = 5 \times 10^{-2}$$ 5. **Express each number in terms of powers of 10:** $$0.2^{32768} = (2 \times 10^{-1})^{32768} = 2^{32768} \times 10^{-32768}$$ $$0.05^{1024} = (5 \times 10^{-2})^{1024} = 5^{1024} \times 10^{-2048}$$ 6. **Compare the orders of magnitude:** - The power of 10 in the first number is $$-32768$$. - The power of 10 in the second number is $$-2048$$. Since $$10^{-32768}$$ is much smaller than $$10^{-2048}$$, the first number is much smaller unless the factors $$2^{32768}$$ and $$5^{1024}$$ compensate. 7. **Compare the factors $$2^{32768}$$ and $$5^{1024}$$:** - Take logarithms base 10: $$\log_{10}(2^{32768}) = 32768 \times \log_{10}(2) \approx 32768 \times 0.30103 = 9868.6$$ $$\log_{10}(5^{1024}) = 1024 \times \log_{10}(5) \approx 1024 \times 0.69897 = 715.9$$ 8. **Calculate the total logarithm of each number:** $$\log_{10}(0.2^{32768}) = 9868.6 - 32768 = -22899.4$$ $$\log_{10}(0.05^{1024}) = 715.9 - 2048 = -1332.1$$ 9. **Interpretation:** Since $$-1332.1 > -22899.4$$, $$0.05^{2^{10}}$$ is larger than $$0.2^{2^{15}}$$. **Final answer:** $$0.05^{2^{10}}$$ is larger than $$0.2^{2^{15}}$$.