Question

1. explain why converting from meters to centimeters uses a different exponent than converting from meters to millimeters.

Explanation:

1. Recall the metric system prefixes: The metric system uses prefixes to represent powers of 10. The prefix "centi - " means $10^{- 2}$, and the prefix "milli - " means $10^{-3}$. But when converting from meters to centimeters or millimeters, we are going from a larger unit to a smaller unit, so we use positive exponents for the conversion factors.

• For converting meters to centimeters: We know that 1 meter is equal to 100 centimeters. In terms of powers of 10, $1\space m=10^{2}\space cm$ (since $100 = 10\times10=10^{2}$). So when converting $x$ meters to centimeters, the formula is $x\space m\times10^{2}\frac{cm}{m}$.
• For converting meters to millimeters: We know that 1 meter is equal to 1000 millimeters. In terms of powers of 10, $1\space m = 10^{3}\space mm$ (since $1000=10\times10\times10 = 10^{3}$). So when converting $y$ meters to millimeters, the formula is $y\space m\times10^{3}\frac{mm}{m}$.

1. Analyze the exponents: The reason the exponents are different is because the relationship between meters and centimeters is based on a factor of $10^{2}$ (since centi - represents $10^{-2}$ in the base unit, but when going from meters to centimeters, we invert the fraction for conversion, so it becomes $10^{2}$), and the relationship between meters and millimeters is based on a factor of $10^{3}$ (since milli - represents $10^{-3}$ in the base unit, and when converting from meters to millimeters, we use the inverse factor of $10^{3}$). The size of the centimeter is larger than the size of the millimeter (1 cm = 10 mm), so the number of centimeters in a meter is less than the number of millimeters in a meter. This difference in the size of the target unit (centimeter vs. millimeter) relative to the meter leads to different powers of 10 (exponents) being used in the conversion.

Answer:

Converting from meters to centimeters uses a factor of $10^{2}$ (since $1\space m = 10^{2}\space cm$) and converting from meters to millimeters uses a factor of $10^{3}$ (since $1\space m=10^{3}\space mm$). The exponents differ because a centimeter is $10$ times larger than a millimeter, so the number of centimeters in a meter ($10^{2}$) is less than the number of millimeters in a meter ($10^{3}$), leading to different powers - of - 10 (exponents) for the conversion factors.