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. \(10^{15}\) Often used to describe processor speed

Short answer

G. \(10^{9}\) is used to describe processor speed as gigahertz (GHz).

Step-by-step solution

Identify the power of 10 used in describing processor speed

Processor speed is often measured in terms of Hertz (Hz). Nowadays, processors operate at gigahertz (GHz) levels, where GHz stands for gigahertz.

Connect gigahertz with powers of 10

The prefix "giga" in gigahertz represents a factor of one billion, or a power of 10 equal to \(10^{9}\). This shows that gigahertz is equivalent to \(10^{9}\) hertz.

Match the power of 10 with the given options

From the list of powers of 10 provided, option G corresponds to \(10^{9}\).

Key concepts

Exponents

Exponents are a foundational mathematical concept used to represent repeated multiplication of a number. They consist of two parts: the base and the exponent itself. The base is the number being multiplied, and the exponent tells you how many times to multiply the base by itself. For example, in the expression \(10^{3}\), 10 is the base, and 3 is the exponent. This means 10 is multiplied by itself three times: \(10 \times 10 \times 10\).Using exponents is incredibly useful in dealing with large numbers. It allows us to express very large or very small quantities succinctly. Understanding exponents is essential in topics like scientific notation, which we'll delve into shortly, and in various fields such as physics and engineering. When you get comfortable with exponents, you'll have a powerful tool for calculating and understanding vast and tiny quantities.

Gigahertz

Gigahertz is a term you might have encountered when reading about electronic devices, especially in the context of computers and smartphones. It is a unit of frequency, where "giga" indicates a billion cycles per second. Therefore, one gigahertz (1 GHz) equals one billion hertz or \(10^{9}\) hertz. In the realm of technology, the speed of a processor is often measured in gigahertz. The higher the gigahertz value, the faster the processor can execute instructions. For example, a processor with a speed of 3 GHz can perform three billion cycles each second! This is essential for determining how swiftly a device can process data, run applications, or perform tasks. Understanding gigahertz helps users assess device capabilities and compare different models based on their processing power.

Processor Speed

Processor speed indicates how quickly a computer or smartphone can execute tasks. It's an essential factor when assessing a device's performance because it directly impacts how efficiently the machine runs software and completes operations. Processor speed is typically expressed in hertz, with modern processors working in terms of gigahertz (GHz). When you see a processor described as having a speed of, say, 2.5 GHz, it's signaling that the processor completes 2.5 billion cycles each second. When choosing a computing device, a higher processor speed is often better, as it allows for faster computations and smoother performance. However, other factors like multi-core processors and RAM also play significant roles in overall performance.

Scientific Notation

Scientific notation is a way of expressing very large or very small numbers using powers of 10. This method is incredibly useful because it simplifies numbers and makes them easier to read and work with, especially in scientific and engineering contexts.To write a number in scientific notation, you move the decimal point until you have a number between 1 and 10, and then multiply by a power of 10 that corresponds to how many places the decimal moved. For example, 3,000 can be written as \(3 \times 10^{3}\). Similarly, 0.0005 can be written as \(5 \times 10^{-4}\).This notation not only saves space and avoids large strings of zeros but also simplifies mathematical operations, making calculations more manageable. Scientific notation is widely used in fields like physics, chemistry, and astronomy, where quantities can be extremely large or minuscule.