What is the value of $[\log_{10} (5 \log_{10} 100)]^2$?

Concept & Formula When dealing with nested mathematical operations, always solve from the innermost parentheses outward, strictly following the order of operations (BODMAS/PEMDAS). $$ \log_a (a^n) = n $$ Step-by-Step Solution * **Given:** Evaluate $[\log_{10} (5 \log_{10} 100)]^2$. * **Calculation:** Start with the innermost term: $\log_{10} 100$. * Since $100 = 10^2$, we know that $\log_{10} 10^2 = 2$. * Substitute this back into the main expression: $$ [\log_{10} (5 \times 2)]^2 $$ * Simplify the term inside the remaining parentheses: $$ [\log_{10} (10)]^2 $$ * Since $\log_{10} 10 = 1$, substitute this value: $$ [1]^2 $$ * Square the result: $$ 1^2 = 1 $$ Exam Strategy & Shortcut This question looks complex but evaluates in seconds mentally. Inner $\log_{10} 100$ is clearly $2$. $5 \times 2$ is $10$. $\log_{10} 10$ is $1$. The square of $1$ is $1$. No pen or paper is needed. Common Pitfall A major mistake is applying the outer square prematurely to the internal elements, or confusing the square bracket with a coefficient (multiplying by $2$ instead of squaring). Always resolve the logs completely before applying outer powers. Final Answer **Therefore, the correct answer is 1.**