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. \(10^{15}\) $$ \text { Pico- } $$

Short answer

The prefix "pico-" corresponds to \(10^{-12}\).

Step-by-step solution

Understanding the Question

The task requires matching the given power of 10 with its corresponding prefix or name that is commonly used in science and engineering, such as pico-, nano-, micro-, etc.

Identify the Pico- Prefix

The prefix "pico-" is used to represent a factor of \(10^{-12}\). This is a standard prefix in the International System of Units (SI) for denoting very small quantities.

Matching the Exponent

The exercise choice corresponding to \(10^{-12}\) is option A. Therefore, \(10^{-12}\) corresponds to the prefix "pico-."

Key concepts

Scientific Notation

Scientific notation is a way to express very large or very small numbers in a more concise form. This format is widely used in scientific and engineering fields. Instead of writing out all the zeros in a number, scientific notation expresses numbers as a product of a coefficient and a power of ten. For example, the number 5,000 can be written as \(5 \times 10^3\).
Scientific notation makes it easier to perform calculations and understand the scale of measurements. When numbers are written in scientific notation, they are all expressed in the same basic format:

• A number greater than or equal to 1 and less than 10.
• Multiplied by a power of 10.

Using scientific notation helps to avoid errors with large or small numbers and simplifies arithmetic operations like addition, subtraction, multiplication, and division. For instance, multiplying \(1.2 \times 10^8\) by \(3 \times 10^4\) simply becomes multiplying the coefficients (1.2 and 3) and then adding the exponents (8 and 4) to get \(3.6 \times 10^{12}\).
Mastering scientific notation is key for algebra, physics, and any field dealing with wide-ranging numerical values.

SI Units

SI Units, or the International System of Units, provide a standard way to express measurements. This system is used globally, which makes scientific data universally interpretable. SI Units cover length, mass, time, electric current, temperature, amount of substance, and luminous intensity.
The foundation of SI Units includes base units like meters (m) for length, kilograms (kg) for mass, and seconds (s) for time. These units combine to form derived units, such as newtons for force or pascals for pressure, which are critical in various scientific calculations.
Understanding and using SI Units ensures precision in measurements and allows for seamless sharing and understanding of data across different countries and disciplines. Whether communicating in science, engineering, or commerce, SI Units help eliminate confusion. Remember, consistency using SI Units is crucial for clarity in scientific work, allowing different researchers to reproduce experiments and verify findings.
Embracing SI Units means you'll be speaking the universal language of science, easing the international collaboration in various fields.

Metric Prefixes

Metric prefixes make it easy to understand the size or magnitude of measurements within the metric system. They are used to indicate specific powers of ten, ranging from very small quantities (like milli- or micro-) to very large ones (like kilo- or mega-).

• The prefix "milli-" corresponds to a factor of \(10^{-3}\), making it useful for expressing small measurements such as millimeters or milliliters.
• The prefix "kilo-" represents \(10^3\) and is commonly seen in kilograms or kilometers.
• Smaller quantities use prefixes like "micro-" \(10^{-6}\) and "nano-" \(10^{-9}\), often used in technology and chemistry fields.

The key to using metric prefixes effectively is understanding their respective powers of ten, allowing for straightforward and clear communication of measurements. This becomes crucial in scientific settings where precision is necessary.
By becoming familiar with metric prefixes, you can easily adapt and understand measurements across a wide range of magnitudes, whether you're dealing with tiny particles or large distances.

Understanding Exponents

Exponents are a powerful mathematical tool used to simplify expressions and calculations involving repeated multiplication of the same number. This is especially true when dealing with powers of ten, crucial in both scientific notation and metric prefixes.
An exponent indicates how many times a number (the base) is multiplied by itself. For example, \(10^3\) means \(10 \times 10 \times 10\). The exponent is 3, showing three multiplications. Exponents can be positive or negative:

• Positive exponents, like \(10^3\), express standard multiplication.
• Negative exponents, such as \(10^{-3}\), represent division (or fractions), essentially the inverse of multiplication. So, \(10^{-3}\) equals \(\frac{1}{10^3}\) or \(0.001\).

Understanding how exponents work allows you to handle vast and minuscule numbers with ease, crucial in scientific fields. They simplify complex calculations, reducing large operations to more manageable steps.
By getting comfortable with exponents, students can adeptly navigate scientific problems, perform calculations faster, and understand the interplay between numerical values in equations.