Problem 65 - College Algebra

Question

Write each number in decimal notation. $$ 4.7 \times 10^{3} $$

Step-by-Step Solution

Answer

The decimal notation of \(4.7 \times 10^{3}\) is 4700

Step 1: Identify the exponent

First, identify the exponent in the scientific notation. Here that is \(3\), which is applied to \(10\). This means that the decimal in \(4.7\) is moved three places to the right.

Step 2: Apply the exponent

To convert the scientific notation to decimal notation, move the decimal point of \(4.7\) right by 3 places as the exponent is \(3\). This leads to a number of \(4700\). If needed, fill in with zeros in the empty decimal places that are created by moving the decimal point.

Key Concepts

Exponent

In scientific notation, the exponent plays a crucial role in determining how the decimal point of a number is adjusted. Think of the exponent as a set of instructions that tells you in which direction and how many places to move the decimal.

• If the exponent is positive, as in our example with 4.7, this signifies that the decimal point shifts to the right. This action increases the value of the number by powers of ten.
• Conversely, a negative exponent would indicate a movement to the left, reducing the number's value.

In the exercise, the exponent is 3. Therefore, we move the decimal point three positions to the right, effectively transforming the number into a larger one.

Decimal Notation

Decimal notation is the traditional way of writing numbers using a base of 10. This is the format most people are familiar with and use in everyday life.

• It involves using digits from 0 to 9 and a decimal point to express fractions of ten.
• In scientific notation exercises like ours, the ultimate goal is to translate the number into decimal notation.

By following the instructions provided by the exponent, we convert the scientific form into a straightforward decimal number. In the given exercise, moving the decimal three places to the right converts 4.7 to 4700.

Conversion Process

The conversion process from scientific notation to decimal notation involves a couple of structured steps. This ensures accurate and easy transformation:
1. **Identify the components**: Recognize the base number (4.7 in our example) and the exponent (3 here).
2. **Move the decimal**: Based on whether the exponent is positive or negative, shift the decimal point accordingly. For a positive exponent like 3, move to the right.
3. **Fill in zeros if necessary**: As you move the decimal, it's common to create empty spaces. Fill these with zeros to maintain numerical accuracy.
By applying these steps, you ensure the efficient conversion of scientific notation numbers to their decimal notation forms. In the example provided, we moved the decimal three places to the right, meeting the 4700 result.