Problem 53 - Algebra 1

Question

SCIENTIFIC NOTATION IN REAL LIFE In Exercises \(53-57\), write the number in scientific notation. LIGHTNING The speed of a lightning bolt is \(120,000,000\) feet per second.

Step-by-Step Solution

Answer

The speed of a lightning bolt can be written in scientific notation as \(1.2 × 10^8\) feet per second.

Step 1: Understand scientific notation

Scientific notation is a way of expressing very large or very small numbers. A number in scientific notation is expressed as \(a × 10^n\), where \(1 ≤ a < 10\) and \(n\) is an integer.

Step 2: Convert the number to scientific notation

The speed of a lightning bolt given in the exercise is \(120,000,000\) feet per second.The number \(120,000,000\) can be written as \(1.2 × 10^8\) in scientific notation. This is because \(120,000,000\) is equivalent to \(1.2\) times \(100,000,000\), or \(1.2\) times \(10^8\).

Step 3: Final answer

So the speed of a lightning bolt, \(120,000,000\) feet per second, can be written in scientific notation as \(1.2 × 10^8\) feet per second.

Key Concepts

Expressing Large Numbers

Expressing large numbers in a simple, compact form is crucial for both scientific and daily life calculations. To avoid writing out numerous zeros, which can be both time-consuming and error-prone, we use scientific notation. Scientific notation allows us to convert large numbers into a product of a number and a power of ten. For instance, the number 120,000,000 becomes much more manageable when expressed as 1.2 × 108.

In our daily lives, we encounter large numbers when talking about distances in astronomy, the national debt, or even the number of cells in the human body. By using scientific notation, we provide a quick and clear understanding of these figures without getting tangled in a web of zeros.

Exponents

An exponent represents the number of times a base number is multiplied by itself. In scientific notation, the exponent is the 'n' in the term a × 10n, where 'a' is a number between 1 and 10, not including 10, and 'n' is an integer. In the context of our lightning bolt example, 8 is the exponent, indicating that the number 10 is multiplied by itself 8 times.

• Positive exponents signify large numbers, e.g., 102 = 100.
• Negative exponents denote fractions or small numbers, e.g., 10-2 = 0.01.

Understanding exponents is essential, as they are key to manipulating scientific notation and can significantly simplify both multiplication and division of large numbers.

Standard Form

In scientific fields, the 'standard form' is synonymous with scientific notation. It's an agreed-upon format for writing very large or very small numbers to maintain uniformity and avoid confusion. For a number to be in standard form, it must be written as a × 10n where 'a' is any number from 1 up to 10, including 1 but not 10, and 'n' is a whole number (positive, negative, or zero).

The standard form is particularly useful when comparing magnitudes of measurements in, say, physics or chemistry, where the numbers can vary exponentially. For example, the charge of an electron or the mass of a galaxy are numbers that are vastly different in size, yet can be expressed neatly in standard form for ease of understanding. This uniform approach simplifies communication and calculations across scientific disciplines.