$0.213 \div 0.00213 = $x$

Concept & Formula When dividing decimals that share the same non-zero digits, the result is always a power of 10. The strategy is to shift the decimal point in both the numerator and denominator by the same number of places until the denominator becomes a whole number. Step-by-Step Solution * Write the division as a fraction: $$\frac{0.213}{0.00213}$$ * Count the decimal places in the denominator: there are 5 decimal places. * To make the denominator a whole number ($213$), multiply both the numerator and the denominator by $100,000$ (which is equivalent to shifting the decimal point 5 places to the right). * Shift the decimal 5 places right in the numerator ($0.213 \rightarrow 21300$). * Shift the decimal 5 places right in the denominator ($0.00213 \rightarrow 213$). * The new fraction is: $$\frac{21300}{213}$$ * Divide the integers: $$\frac{21300}{213} = 100$$ Exam Strategy & Shortcut Just count the decimal places. The numerator has 3, and the denominator has 5. Use the exponent rule: $10^{-3} / 10^{-5} = 10^{-3 - (-5)} = 10^2 = 100$. Since the significant digits ($213$) are identical, they cancel out to $1$, leaving only the power of 10. Common Pitfall A common error is shifting the decimal point in the wrong direction or shifting it by the numerator's count (3 places) instead of the denominator's count, resulting in $\frac{213}{2.13}$, which still leaves a decimal division to perform and invites further mistakes. Final Answer **Therefore, the correct answer is 100.**