Problem 63 - Introductory Algebra for College Students

Question

Write 0.00397 in scientific notation.

Step-by-Step Solution

Answer

The scientific notation for 0.00397 is \(3.97 \times 10^{-3}\).

Step 1: Finding the Base

First, locate the decimal to make the base number between 1 and 10. For 0.00397, the number that would fall between 1 and 10 is 3.97.

Step 2: Counting the Decimal Places

Recall that in scientific notation, numbers are expressed as b x 10^n. Count how many places the decimal would move to go from 0.00397 to 3.97. In this case, the decimal moves 3 places.

Step 3: Determining the Sign of Exponent

The exponent is positive if the decimal is moved to the left and negative if it is moved to the right. Since you moved the decimal to the right, the exponent is -3.

Step 4: Expressing the number in Scientific Notation

Having determined b=3.97 and n=-3, express the number in scientific notation as \(3.97 \times 10^{-3}\).

Key Concepts

Base Number

In scientific notation, the base number is a key component. It represents the "significant" digits of the original number, but put into a form that falls between 1 and 10. It is achieved by moving the decimal point in the original number to a position where only one non-zero digit is in front of the decimal point.
For example, in the number 0.00397, the base number becomes 3.97. This is because moving the decimal three places to the right places it appropriately between 1 and 10.
Remember, finding the base number is the first step in transforming standard numbers into scientific notation.

Decimal Places

Decimal places play an important role in converting a number to scientific notation. They determine how much you need to move the decimal point to transform the number. The number of positions you move the decimal point becomes the exponent in the scientific notation.
In 0.00397, the decimal point moves three places to the right to become 3.97. This movement is crucial because it directly influences what the exponent in the scientific notation will be.

Exponent Sign

The exponent sign is determined by the direction in which you move the decimal point to transform the original number into the base.
Here are the rules:

• If you move the decimal to the right, like in 0.00397 to 3.97, the exponent is negative.
• If you move it to the left, the exponent is positive.

The exponent tells us how many places the decimal was moved and in what direction. For our example number, since the decimal moved 3 places to the right, the exponent becomes -3.

Expressing Numbers in Scientific Notation

Once you've identified your base number and figured out your exponent, you can write the number in scientific notation format. This format is very useful for writing extremely large or small numbers in a concise way. The expression follows the format:

• Base number (b) multiplied by 10 raised to the power of the exponent (n): \(b \times 10^n\)

In our example, the base number was 3.97 and the exponent was -3, resulting in 3.97 \(\times 10^{-3}\). This notation makes it efficient to process and understand diminutive numbers like 0.00397.