Question

For Exercises \(1-16\), match the power of 10 to its name or use. A. \(10^{-12}\) B. \(10^{-9}\) C. \(10^{-6}\) D. \(10^{-3}\) E. \(10^{3}\) F. \(10^{6}\) G. \(10^{9}\) H. \(10^{12}\) I. \(10^{15}\) $$ \text { Nano- } $$

Short answer

Option B: \(10^{-9}\) is 'Nano-'.

Step-by-step solution

Understanding Prefixes

The word 'Nano-' refers to a unit prefix in the metric system denoting a factor of \(10^{-9}\). This prefix is commonly used in scientific measurements to indicate a billionth of a unit.

Matching the Power of 10

Based on the known relationship that 'Nano-' corresponds to \(10^{-9}\), we match the given options to find the correct power of 10. The option with \(10^{-9}\) is B.

Key concepts

Scientific Notation

Scientific notation is a method used to express very large or very small numbers concisely. It is particularly useful in science and engineering, where precision with large figures is necessary. In scientific notation, a number is written as the product of two factors:

• A coefficient that is greater than or equal to 1 and less than 10.
• A power of ten that indicates the factor by which the coefficient should be multiplied.

For example, the scientific notation for the number 6500 is written as \(6.5 \times 10^3\). Here, 6.5 is the coefficient, and \(10^3\) is the power of ten, indicating a multiplication by 1000.
When dealing with very small numbers, scientific notation provides clarity. For instance, 0.000001 can be expressed as \(1 \times 10^{-6}\), making the scale of the number more apparent. This method facilitates easier reading and comparison of values across varied scales.

Measurement Units

Measurement units play a crucial role in accurately describing quantities in various fields such as science, engineering, and daily life. Units like meters, grams, and seconds form the base of most measurement systems. However, to express quantities efficiently, particularly those that are enormously large or tiny, we use metric prefixes.

• Metric prefixes like 'kilo-', 'mega-', 'giga-', 'milli-', and 'micro-' help scale these base units by powers of ten.
• The prefix 'nano-' is equal to \(10^{-9}\), representing one-billionth of the base unit.

For example, a nanometer is \(10^{-9}\) meters, which is important in fields like nanotechnology that deal with extremely small scales.
Understanding metric prefixes helps in converting measurement units and making sense of various physical quantities. It's an integral part of both basic and advanced scientific studies.

Powers of Ten

Powers of ten are fundamental in mathematical calculations and representations, offering a simple way to scale numbers up or down. A power of ten is an expression involving the number ten raised to an exponent. The exponent tells us how many times to use ten in a multiplication.For instance:

• \(10^3 = 1000\), indicating three tens multiplied together.
• \(10^{-3} = 0.001\), representing the reciprocal of \(10^3\).

These powers form the basis for scientific notation and are employed in expressing measurement units using metric prefixes.
When dealing with powers of ten, it is essential to understand that the sign of the exponent indicates direction:

• A positive exponent denotes multiplication and thus a larger number.
• A negative exponent indicates division and results in a smaller number.

This concept makes calculations simpler and ensures accuracy, particularly with very large or very small numbers.