Question

Convert the following expressions into exponential notation: (a) 3 terameters \((\mathrm{tm})\) (b) 2.5 femtoseconds (fs) (c) 57 micrometers \((\mu m)\) (d) 8.3 megagrams (mg).

Short answer

(a) 3 Terameters (Tm) = \(3 \times 10^{12}\) meters (m) (b) 2.5 Femtoseconds (fs) = \(2.5 \times 10^{-15}\) seconds (s) (c) 57 Micrometers (μm) = \(57 \times 10^{-6}\) meters (m) (d) 8.3 Megagrams (Mg) = \(8.3 \times 10^{6}\) grams (g)

Step-by-step solution

Understating the prefixes

The given prefixes have the following exponential values: - Tera (T) means 10^12 - Femto (f) means 10^(-15) - Micro (μ) means 10^(-6) - Mega (M) means 10^6 Now, we will convert the quantities.

(a) Converting 3 terameters to exponential notation

As Tera (T) means 10^12, we can replace the T with the exponential notation: 3 Terameters (Tm) = \(3 \times 10^{12}\) meters (m)

(b) Converting 2.5 femtoseconds to exponential notation

As Femto (f) means 10^(-15), we can replace the f with the exponential notation: 2.5 Femtoseconds (fs) = \(2.5 \times 10^{-15}\) seconds (s)

(c) Converting 57 micrometers to exponential notation

As Micro (μ) means 10^(-6), we can replace the μ with the exponential notation: 57 Micrometers (μm) = \(57 \times 10^{-6}\) meters (m)

(d) Converting 8.3 megagrams to exponential notation

As Mega (M) means 10^6, we can replace the M with the exponential notation: 8.3 Megagrams (Mg) = \(8.3 \times 10^{6}\) grams (g)

Key concepts

Metric Prefixes

Metric prefixes are essential in handling large or small quantities in measurements efficiently. These prefixes represent powers of ten and are attached to the base unit to indicate the magnitude of the value. Some commonly used metric prefixes include:

• Tera (T): Represents a factor of \(10^{12}\). It's used for extremely large quantities, like terameters or terabytes.
• Femto (f): Stands for \(10^{-15}\). It helps in denoting very tiny amounts, often used in scientific research, such as femtoseconds.
• Micro (μ): Implicates \(10^{-6}\), typically used in fields such as biology and electronics, like micrometers or micrograms.
• Mega (M): Equates to \(10^{6}\). It's useful in various industries when dealing with millions, like megabytes and megagrams.

Understanding these prefixes is crucial when performing conversions in science and engineering to ensure accurate calculations and communication.

Scientific Notation

Scientific notation is a method used to express very large or very small numbers in a compact form. This notation makes it easier to read, write, and communicate these numbers, especially in scientific and engineering contexts. In scientific notation, a number is expressed in the form of \(a \times 10^{n}\), where \(a\) is a coefficient that is greater than or equal to 1 and less than 10, and \(n\) is an integer.

For instance, the number 3000 can be written as \(3 \times 10^{3}\), and 0.00025 can be expressed as \(2.5 \times 10^{-4}\).

• This method helps in managing numbers that belong to both ends of the scale, such as the distance between stars and the size of atoms.
• Scientific notation also aids in performing mathematical operations like multiplication and division, as it allows easy handling of powers of ten.

By transforming quantities using exponential notation, scientific notation ensures that information can be communicated efficiently and clearly.

Unit Conversion

Unit conversion is the process of changing a measurement from one unit to another. This process is often essential in science, engineering, and everyday life when needing to compare or integrate measurements from different systems. To perform a unit conversion, you follow these general steps:

• Identify the original unit and the desired unit: Understanding what you have and what you need.
• Use a conversion factor: A conversion factor is a ratio that expresses how many of one unit are equal to another. For example, there are 1000 milliliters in a liter, so the conversion factor is \(1000 \text{ ml/l}\).
• Multiply the original measurement by the conversion factor: This will give you the measurement in the desired unit. Consistent units make it easier to perform further calculations or comparisons.

It's important to be precise with unit conversions to avoid errors, especially in technical fields where precision is crucial. Understanding unit conversion fundamentals allows quick and accurate translations between different measurement systems, enhancing communication and efficiency.