The number of digits in $4^9 \times 5^{17}$, when expressed in usual form, is

### Concept & Strategy While this could theoretically be solved using logarithms, the most efficient method is basic exponent manipulation. By converting the bases into $2$ and $5$, we can pair them up to form powers of $10$. Every factor of $10$ adds exactly one trailing zero to the final number. ### Step-by-Step Solution Given the expression: $$ 4^9 \times 5^{17} $$ Step 1: Convert the base $4$ into base $2$ so it can interact with the base $5$. $$ (2^2)^9 \times 5^{17} $$ Step 2: Multiply the inner and outer exponents for the base $2$. $$ 2^{18} \times 5^{17} $$ Step 3: To pair the $2$s and $5$s into $10$s, match their exponents. We can split $2^{18}$ into $2^1 \times 2^{17}$. $$ 2^1 \times 2^{17} \times 5^{17} $$ Step 4: Combine the matched powers into a power of $10$. $$ 2 \times (2 \times 5)^{17} $$ $$ 2 \times 10^{17} $$ Step 5: Count the digits. The expression $2 \times 10^{17}$ translates to the digit "2" followed by exactly $17$ zeros. $$ \text{Total digits} = 1 \text{ (for the digit 2)} + 17 \text{ (zeros)} = 18 $$ ### Exam Strategy & Shortcut **Zero Pairing:** Recognize instantly that $4^9$ is $2^{18}$. You have eighteen $2$s and seventeen $5$s. You can form exactly seventeen $10$s (giving you $17$ zeros). You are left with one stray $2$. A single digit number multiplied by $10^{17}$ will always have exactly $18$ digits. This bypasses logarithmic calculations entirely. ### Common Pitfall Attempting to solve this by finding $\log(4^9) + \log(5^{17})$ manually without a calculator is a massive time sink. While mathematically sound, multiplying $\log 4$ by $9$ and $\log 5$ by $17$ introduces numerous opportunities for arithmetic mistakes compared to basic exponent grouping. ### Final Answer **Therefore, the correct answer is 18.**