Question

What metric prefixes correspond to the following powers of \(10 ?\) a) \(10^{3}\) b) \(10^{-2}\) c) \(10^{-3}\)

Short answer

a) \(10^{3}\) b) \(10^{-2}\) c) \(10^{-3}\) Answer: a) kilo (k) b) centi (c) c) milli (m)

Step-by-step solution

Recall the metric prefixes and their corresponding powers of 10

The most common metric prefixes are: - yotta (Y): \(10^{24}\) - zetta (Z): \(10^{21}\) - exa (E): \(10^{18}\) - peta (P): \(10^{15}\) - tera (T): \(10^{12}\) - giga (G): \(10^{9}\) - mega (M): \(10^{6}\) - kilo (k): \(10^{3}\) - hecto (h): \(10^{2}\) - deka (da): \(10^{1}\) - deci (d): \(10^{-1}\) - centi (c): \(10^{-2}\) - milli (m): \(10^{-3}\) - micro (µ): \(10^{-6}\) - nano (n): \(10^{-9}\) - pico (p): \(10^{-12}\) - femto (f): \(10^{-15}\) - atto (a): \(10^{-18}\) - zepto (z): \(10^{-21}\) - yocto (y): \(10^{-24}\)

Match the powers of 10 to their metric prefixes

a) \(10^{3}\) corresponds to the kilo (k) prefix. b) \(10^{-2}\) corresponds to the centi (c) prefix. c) \(10^{-3}\) corresponds to the milli (m) prefix.

Key concepts

Powers of Ten

Understanding the powers of ten is foundational to grasping various scientific and mathematical concepts, including metric prefixes. At its core, a power of ten simply means using ten to the power of a number. For example, when we say \(10^3\), we mean ten multiplied by itself three times: \(10 \times 10 \times 10 = 1000\). This is a way to express large or small numbers compactly and to simplify calculations and comparisons.

In the context of the exercise, recognizing that \(10^3\), \(10^{-2}\), and \(10^{-3}\) represent 1000, 0.01, and 0.001, respectively, helps us link these values to metric prefixes. It's essential for students not just to memorize these values, but to understand that each increment in the exponent corresponds to the number being multiplied or divided by ten.

This concept becomes extremely practical when dealing with units of measurement or converting between scales in science and engineering. After all, being fluent with the powers of ten is like having a mental calculator for processing the scale of numbers efficiently.

Scientific Notation

Scientific notation is a method of writing numbers that accommodates values too large or too small to be conveniently written in decimal form. It's also called standard form. For instance, instead of writing out 0.000000123, you could write \(1.23 \times 10^{-7}\), which is much easier to read and comprehend. In scientific notation, a number is written as a product of two parts: a coefficient (usually between 1 and 10) and a power of ten.

Looking at the provided exercise, using scientific notation simplifies the understanding of quantities and their corresponding metric prefixes. The exercise's objective—identifying metric prefixes for certain powers of ten—is ideally suited for students to appreciate scientific notation. A major benefit of this notation is that it makes it easier to perform operations like multiplication and division on very large or very small numbers because the powers of ten can be quickly added or subtracted.

SI Units

SI units, known as the International System of Units, provide a standardized system of measurements used globally for science, industry, and commerce. These include meters (m) for length, kilograms (kg) for mass, seconds (s) for time, and so on. At the heart of the system are the seven base units from which other units can be derived. Crucially, the SI system is based on metric prefixes, which essentially act as multipliers of the base unit.

As each prefix represents a power of ten, we can convert between units easily. For example, 1 kilometer (km) is 1,000 meters (m) because 'kilo-' corresponds to \(10^3\), or 1,000 times the base unit of a meter. Similarly, 1 milligram (mg) is 0.001 grams (g) because 'milli-' corresponds to \(10^{-3}\). The exercise demonstrates this utility by associating specific powers of ten with their metric prefixes, integrating seamlessly with the SI unit system. Mastering these prefixes means navigating the landscape of measurements with confidence, crucial for any budding scientist or engineer.