Problem 13 - General Chemistry: Principles and Modern Applications

Question

Express each number in exponential notation. (a) \(8950 .;\)(b) \(10,700 . ;(\text { c) } 0.0240 ; \text { (d) } 0.0047 ; \text { (e) } 938.3 ; \text { (f) } 275,482\).

Step-by-Step Solution

Answer

(a) \(8.95 * 10^{3}\) (b) \(1.07 * 10^{4}\) (c) \(2.40 * 10^{-2}\) (d) \(4.7 * 10^{-3}\) (e) \(9.383 * 10^{2}\) (f) \(2.75482 * 10^{5}\)

Step 1: Expressing numbers in exponential notation

The rule to express numbers in exponential notation is to have one non-zero digit to the left of the decimal. Move the decimal point as needed to create this number and then record the number of places and direction you moved the decimal as a power of 10. If the decimal was moved to the left, the power of 10 will be positive and if it was moved to the right, the power of 10 will be negative.

Step 2: Apply the rule to the given numbers

(a) For the number 8950, move the decimal three places to the left. The number is then written as \(8.95 * 10^{3}\).\n (b) For the number 10,700, move the decimal four places to the left. The number is then written as \(1.07 * 10^{4}\).\n (c) For the number 0.0240, move the decimal two places to the right. The number is then written as \(2.40 * 10^{-2}\).\n (d) For the number 0.0047, move the decimal three places to the right. The number is then written as \(4.7 * 10^{-3}\).\n (e) For the number 938.3, move the decimal two places to the left. The number is then written as \(9.383 * 10^{2}\).\n (f) For the number 275,482, move the decimal five places to the left. The number is then written as \(2.75482 * 10^{5}\).

Key Concepts

Scientific Notation

Scientific notation is a way of expressing very large or very small numbers in a concise form. It consists of two parts:

• A coefficient, which is a number usually between 1 and 10.
• And a power of ten, which indicates how many places the decimal point has been moved.

This method of notation helps to standardize the format and makes calculations easier. For example, the number 8950 expressed in scientific notation is written as \(8.95 \times 10^{3}\).
Here, the "8.95" is the coefficient and "\(10^{3}\)" is the power of ten indicating that the decimal was moved three places to the left. When converting a number into scientific notation, remember:

• The coefficient should have only one non-zero digit to the left of the decimal.
• Adjust the power of ten based on how many places the decimal moves.
• A positive exponent is used when you move the decimal to the left.
• A negative exponent is used when the decimal is moved to the right.

Powers of Ten

Understanding how powers of ten work is crucial for working with scientific notation. Powers of ten are simply ten multiplied by itself a certain number of times. Here are a few examples:

• \(10^{1} = 10\)
• \(10^{2} = 100\)
• \(10^{3} = 1000\)

These examples show that the exponent (or the power) signifies how many zeros follow the number 1. For smaller numbers, the decimal point notation becomes important:

• \(10^{-1} = 0.1\)
• \(10^{-2} = 0.01\)
• \(10^{-3} = 0.001\)

Negative powers indicate that we are dealing with fractions of ten, or decimals. Recognizing how powers of ten work will help you understand why a number like 0.0047 converts to \(4.7 \times 10^{-3}\), because moving the decimal three places to the right increases the exponent by \(-3\).

Decimal Point

The decimal point is the small dot used to separate the integer part from the fractional part of a number. Understanding how to manipulate the decimal point is vital when dealing with scientific and exponential notation. Let's take a closer look at its role:

• For whole numbers, the decimal point is placed after the last digit, although it is often omitted. For example, 275482 can be rewritten as 275482.0.
• Moving the decimal point to the left increases the power of ten. For example, moving the decimal in 275482 five places to the left makes it 2.75482 and gives us a power of \(10^{5}\).
• For numbers less than one, we move the decimal point to the right to create a coefficient between 1 and 10, thus decreasing the power of ten, leading to a negative exponent. For instance, 0.0240 becomes 2.40 when you shift the decimal two places to the right, equating to \(10^{-2}\).

Manipulating the decimal point carefully and correctly is the key to mastering scientific notation. Each movement translates directly into the power term of the notation, ensuring accuracy in representing and calculating numbers.