Which of the following is closest to zero?

### Concept & Logic To find the value closest to zero, evaluate each mathematical expression carefully. Squaring a decimal fraction between $0$ and $1$ yields an even smaller decimal fraction. ### Step-by-step Solution Let's calculate the exact value for each option: * **Option (a):** $(0.09)^2$ $$0.09 \times 0.09 = 0.0081$$ * **Option (b):** $0.09$ This is simply $0.0900$. * **Option (c):** $(1 - 0.9)^2$ First, solve inside the parenthesis: $(0.1)^2$ $$0.1 \times 0.1 = 0.0100$$ * **Option (d):** $1 - (0.9)^2$ First square $0.9$: $(0.9)^2 = 0.81$ Then subtract: $1 - 0.81 = 0.1900$ Comparing the evaluated numbers: $0.0081$, $0.0900$, $0.0100$, and $0.1900$. The smallest magnitude (closest to absolute zero) is $0.0081$. ### Exam Strategy & Shortcut Recognize that squaring a very small decimal makes it exponentially smaller. Comparing $(0.09)^2$ with $(0.1)^2$ from option (c), we immediately know that $(0.09)^2$ will be the absolute smallest without executing the full calculation. ### Common Pitfall Misplacing the decimal point when squaring. Many students mistakenly calculate $(0.09)^2$ as $0.081$ instead of $0.0081$. Remember to count the total decimal places: two in $0.09$ plus two in $0.09$ equals four decimal places in the product. ### Final Answer Therefore, the correct answer is $(0.09)^2$.