1. The problem asks for the number of digits in the number $8^{10} \times 5^{22}$. 2. To find the number of digits of a positive integer $N$, we use the formula: number of digits = $\lfloor \log_{10}(N) \rfloor + 1$. 3. First, rewrite the expression: $$8^{10} \times 5^{22} = (2^3)^{10} \times 5^{22} = 2^{30} \times 5^{22}$$ 4. Separate powers of 2 and 5: $$2^{30} \times 5^{22} = 2^{22} \times 2^{8} \times 5^{22} = (2^{22} \times 5^{22}) \times 2^{8} = 10^{22} \times 2^{8}$$ 5. Calculate $2^{8}$: $$2^{8} = 256$$ 6. So the number is: $$10^{22} \times 256 = 256 \times 10^{22}$$ 7. This is a 256 followed by 22 zeros. 8. The number of digits is the digits in 256 plus 22 zeros: $$3 + 22 = 25$$ **Final answer:** The number $8^{10} \times 5^{22}$ has 25 digits.