Problem 77 - College Algebra

Question

In Exercises 77–86, write each number in scientifi c notation. 32,000

Step-by-Step Solution

Answer

The scientific notation of 32,000 is \(3.2 \times 10^4\).

Step 1: Understanding Scientific Notation

Scientific notation is a way of expressing numbers that are too big or too small to be conveniently written in standard decimal form. It is of the form \(a \times 10^n\), where \(1 \leq |a| < 10\) and \(n\) represents the number of places the decimal point has been moved.

Step 2: Convert the Number into Scientific Notation

The number is 32,000. We can express it in the form \(a \times 10^n\). The number 32,000 can be written as 3.2 multiplied by 10,000 which needs 4 places to move the decimal point. Thus, the scientific notation will be \(3.2 \times 10^4\).

Key Concepts

Standard Decimal Form

Standard decimal form is a way to write numbers using a base of 10 and allowing for the use of decimal points and digits. It is the form we most commonly use in everyday arithmetic. When a number is written in standard decimal form, it can be a whole number, a decimal, or a combination of both. This standard form makes it easy for us to understand and compute different mathematical operations.

Here’s what you should keep in mind about the standard decimal form:

• It represents numbers with digits, using the base of 10.
• Decimal points are used to separate the integer part from fractional parts.
• It is straightforward but can become cumbersome for very large or very small numbers.

Large numbers, like 32,000, are often unwieldy in standard decimal form, leading to potential mistakes or oversights, especially when performing multiple operations. This is why scientific notation becomes useful, as it simplifies how these numbers are expressed.

Decimal Point

The decimal point is a crucial element in the representation of numbers in a more precise manner. It serves as the separator between the whole number and the fractional part of a number. When expressing numbers, its position can determine the value of each digit in the number.

Some key points about the decimal point include:

• The digits to the left of the decimal point make up the whole number.
• The digits to the right of the decimal point are fractional, diminishing in value as they move further right.
• Moving the decimal point to the left or right changes a number's value significantly.

In scientific notation, the decimal point’s placement is essential. For instance, when writing the number 32,000 in scientific notation, the decimal point is shifted four places to the right of the number 3.2. Thus, the number becomes simpler to express and understand.

Expressing Numbers

Expressing numbers accurately and efficiently can vary depending on the context and the size of the number. For extremely large or tiny numbers, using scientific notation makes the process practical and reduces the chance for error.

Expressing numbers using scientific notation involves:

• Identifying a significant figure or coefficient, which is a number between 1 and 10.
• Determining the exponent of 10, which indicates how many places and in which direction the decimal point has to move to revert to standard decimal form.
• Using the format: \( a \times 10^n \), where \( a \) is the significant figure and \( n \) is the power of ten.

By converting 32,000 into the scientific notation of \( 3.2 \times 10^4 \), the focus is shifted from counting zeros to understanding the magnitude of the number. This conversion makes it easier to handle and compute with both extremely large and small numbers.