Problem 3 - Astronomy: At Play in the Cosmos

Question

Which of the following is equivalent to \(9,400,000 ?\) a. \(9.4 \times 10^{5}\) d. \(0.94 \times 10^{8}\) b. \(9.4 \times 10^{6}\) e. \(0.94 \times 10^{9}\) c. \(9.4 \times 10^{7}\)

Step-by-Step Solution

Answer

The equivalent expression is \(9.4 \times 10^{6}\) (Option b).

Step 1: Understand What Is Being Asked

We need to find which of the given options is equivalent to the number 9,400,000 written in scientific notation.

Step 2: Recall the Definition of Scientific Notation

Scientific notation is a way of writing numbers that are too big or too small to be conveniently written in decimal form. A number is written as \(a \times 10^{n}\), where \(1 \le a < 10\) and \(n\) is an integer.

Step 3: Convert 9,400,000 to Scientific Notation

Write 9,400,000 in the form \(a \times 10^{n}\). We can express it as \(9.4 \times 10^{6}\) because moving the decimal point 6 places to the left places the decimal immediately after the 9.

Step 4: Select the Correct Answer

Compare the expression \(9.4 \times 10^{6}\) with the given options. Option b, \(9.4 \times 10^{6}\), is the correct match.

Key Concepts

Exponentiation

Exponentiation is a fundamental mathematical operation. It involves raising a number, known as the base, to the power of an exponent. This tells us how many times we multiply the base by itself. For example, in the expression \(10^6\), 10 is the base, and 6 is the exponent. This means you multiply 10 by itself six times:

• \(10 \times 10 \times 10 \times 10 \times 10 \times 10 = 1,000,000\)

Exponentiation is useful when dealing with very large or very small numbers, as it provides a concise way to express them. It's a key component of scientific notation used in the exercise.
The exponent tells you the position of the decimal point, moving it one place for each order of magnitude. In scientific notation, this is vital, as it reflects the scale of the number succinctly.

Decimal Notation

Decimal notation is the standard way of writing numbers using digits 0-9. It's familiar to most and is used in everyday life for anything involving numbers. For instance, 9,400,000 is a number that appears in decimal notation. Simply put, decimal notation arranges these digits in orders of magnitude using place values like ones, tens, hundreds, thousands, and so on.
Unlike scientific notation, decimal notation can be cumbersome for very large or tiny numbers. Writing out 9,400,000 fully takes up a lot of space and is not as convenient as its scientific form. That's why methods like scientific notation are preferred in science and engineering when calculations involve really large or small numbers.

Scientific Method

The scientific method here refers to the systematic approach of using scientific notation for handling numbers that are otherwise difficult to write in decimal form. This method follows a straightforward format: \(a \times 10^{n}\).

• \(a\) must be a number between 1 and 10 (not including 10 itself), ensuring easy comparison and comprehension.
• \(n\) represents how many times the base (10) is multiplied by itself, dictating the movement of the decimal point.

To convert a number like 9,400,000 into scientific notation, determine where to position the decimal (after the first significant figure, making \(a = 9.4\)), and count how many places it moves to reach this form. Here it moves six places to the left, resulting in \(9.4 \times 10^6\). This systematic approach aids in expressing huge numbers efficiently, reducing complexity and potential errors, and is invaluable in scientific contexts.