1. **State the problem:** Find the number of digits in the number $5^{22} \times 8^{10}$. 2. **Recall the formula for the number of digits:** The number of digits of a positive integer $N$ is given by $\lfloor \log_{10}(N) \rfloor + 1$. 3. **Rewrite the expression:** $$5^{22} \times 8^{10} = 5^{22} \times (2^3)^{10} = 5^{22} \times 2^{30}$$ 4. **Combine powers of 2 and 5:** $$5^{22} \times 2^{30} = 5^{22} \times 2^{22} \times 2^{8} = (5 \times 2)^{22} \times 2^{8} = 10^{22} \times 2^{8}$$ 5. **Calculate the number of digits:** The number is $10^{22} \times 2^{8} = 10^{22} \times 256$. 6. **Number of digits of $10^{22} \times 256$:** Since $10^{22}$ is a 1 followed by 22 zeros (23 digits), multiplying by 256 shifts the digits. Calculate $\log_{10}(10^{22} \times 256) = \log_{10}(10^{22}) + \log_{10}(256) = 22 + \log_{10}(256)$. Calculate $\log_{10}(256)$: $$256 = 2^8 \Rightarrow \log_{10}(256) = 8 \log_{10}(2) \approx 8 \times 0.30103 = 2.40824$$ So, $$\log_{10}(10^{22} \times 256) = 22 + 2.40824 = 24.40824$$ Number of digits = $\lfloor 24.40824 \rfloor + 1 = 24 + 1 = 25$. **Final answer:** The number $5^{22} \times 8^{10}$ has **25 digits**.