Question

SCIENTIFIC NOTATION Rewrite in scientific notation. $$ 0.0000288 $$

Short answer

The number 0.0000288 in scientific notation is \(2.88 \times 10^{-5}\)

Step-by-step solution

Identify the position

Identify the first non-zero number, ideally between 1 and 10 (in this case 2). So the decimal number 0.0000288 will have the number 2.88 in scientific notation.

Count the decimal places

Count the decimal places to be moved. The decimal point will move 5 places to the right. The number of places, counted from the original position of the decimal point, will be the exponent of 10.

Establish the scientific notation

Establish the scientific notation, that is, multiply the identified number by 10 to the power of the count. The equation becomes \(2.88 \times 10^{-5}\)

Key concepts

Converting Decimals to Scientific Notation

Converting a decimal number to scientific notation is a useful technique for simplifying the representation of very large or very small numbers. The goal is to rewrite the number so that it’s easier to handle in calculations, especially in fields such as physics, engineering, and chemistry.

To convert a decimal to scientific notation:

• Start by identifying the first non-zero digit in the number.
• Place a decimal point after this digit to form a new number between 1 and 10.
• Count how many places you move the decimal point to get from the original number to this new number.
• Multiply this new number by 10 raised to the power of the number of places moved. If you moved the decimal point to the left, this exponent is positive. If you moved it to the right, the exponent is negative.

Real-life Application

Imagine you are working in a laboratory, and you need to measure substances with very small weights or in very high quantities. Using scientific notation allows you to avoid errors and easily compare different measurements by providing a standardized format for these values.

Exponents in Scientific Notation

In scientific notation, the exponent represents the number of places the decimal point has been moved to transform the original number into a new number between 1 and 10. The exponent is an essential part of scientific notation, as it indicates the magnitude of the number.

Important points about exponents in scientific notation include:

• The exponent is positive if the original decimal was moved to the left because the original number is greater than 10.
• The exponent is negative if the decimal was moved to the right since the original number is less than one.
• The exponent is written as a superscript to the right of the base number 10.
• The number 10 is used as the base because it reflects the decimal (base 10) system.

Why the Base 10?

The base 10 makes calculations consistent with the decimal system and simplifies the multiplication or division of numbers with different magnitudes.

Scientific Notation Steps

To master the process of writing numbers in scientific notation, follow this structured approach:

1. Identify the non-zero digits in the original decimal number.
2. Determine the appropriate position for the new decimal point such that the resulting number falls between 1 and 10, which is the standard form.
3. Count the number of places the decimal point moves to reach this position; this count will be your exponent.
4. Construct the scientific notation by combining the newly formed number and 10 raised to the power of your count (use a negative exponent if the decimal moves to the right).

Practice Example

Consider the number 0.0000288. Here, you would move the decimal point 5 places to the right (eastward on the number line!), landing you with 2.88, and since the move was to the right, the exponent will be negative. Thus, after applying the aforementioned steps, the number in scientific notation is expressed as: \(2.88 \times 10^{-5}\). Practice with a variety of numbers will build confidence and fluency in using scientific notation.