Problem 57 - College Algebra and Calculus: An Applied Approach

Question

Write the number in scientific notation.Light Year: \(9,460,000,000,000\) kilometers

Step-by-Step Solution

Answer

The number \(9,460,000,000,000\) in scientific notation is \(9.46 \times 10^{13}\)

Step 1: Understanding Scientific Notation

Scientific notation is a way to express large or small numbers in a more readable format. It is typically in the form \(a \times 10^n\), where \(a\) is a number between 1 and 10 (including 1 but excluding 10) and \(n\) is an integer.

Step 2: Preparing the Number

In our case, the given number is \(9,460,000,000,000\). Starting from the leftmost number (which is the number 9), count the number of places or digits until you come to the end of the number. Here, there are 13 places or digits which is the value of our exponent, \(n\).

Step 3: Converting into Scientific Notation

In scientific notation, we write this number as \(9.46\), which is obtained by moving the decimal point to the right of the first digit (in this case, the 9) and adjusting the exponent accordingly. We know from the previous step that the exponent is 13, so the complete scientific notation for this number is \(9.46 \times 10^{13}\)

Key Concepts

Large Numbers

Large numbers can be overwhelming. Imagine trying to keep track of something like 9,460,000,000,000! But, understanding them becomes easier with scientific notation.
Large numbers, like a light year in kilometers, are quite common in science. Using traditional representation is impractical due to the frequency of repetition (e.g., counting zeros) and potential for error.

Scientific notation simplifies these numbers, making them manageable and easier to work with in calculations. It reduces the risk of mistakes while improving readability in scientific documents and calculations.

• In our example, the number 9,460,000,000,000 is large and cumbersome to handle in standard form.
• Scientific notation condenses it into a compact form: 9.46 × 10¹³.

Recognizing large numbers and converting them accurately using scientific notation is a skill that takes practice. It is essential in fields like astronomy, physics, and engineering.

Exponents

Exponents are a way of expressing repeated multiplication. They are integral to the concept of scientific notation, as they define the magnitude of the number.

An exponent consists of a base (often 10 in scientific notation) and a power, which shows how many times the base is multiplied by itself. For instance, in 10², 10 is multiplied by itself twice to get 100.

• In scientific notation, exponents tell us how to shift the decimal point.
• Using exponents simplifies calculations, especially with very large or very small numbers.

In the case of the light year example, the exponent is 13. When you move the decimal 13 places to convert 9,460,000,000,000 into 9.46, the exponent 13 signifies those shifts to the right, indicating that the original number is 9.46 followed by thirteen zeros.

Decimal Point

The decimal point is crucial in scientific notation. It determines how numbers are formatted, particularly when dealing with very large or very small values in science.

When converting a large number into scientific notation, the position of the decimal point is adjusted.
In standard form, the number may not even have a visible decimal point, but it's understood to be at the end of the number if there is no fraction.

• In 9,460,000,000,000, the decimal is after the last zero.
• To convert, move the decimal left towards the first non-zero digit.

The goal is to make the first number a single digit between 1 and 10. For our light year example, moving the decimal 13 places to the left positions it right after the 9, resulting in 9.46.
This precise placement of the decimal influences the exponent and assures that the transformation into scientific notation is accurate.