Problem 130 - College Algebra Essentials

Question

Explain how to convert from decimal to scientific notation and give an example.

Step-by-Step Solution

Answer

To convert 34500 into scientific notation, rewrite the number as a decimal between 1 and 10. Then, multiply this number by 10 raised to the power of the number of places the decimal point was moved. This gives \(3.4500 × 10^4\).

Step 1: Understanding Scientific Notation

Scientific notation is a way of writing numbers that are too large or too small to be conveniently written as decimals. In scientific notation, a number is represented as the product of a number between 1 and 10 (inclusive) and a power of 10.

Step 1: Identify the given decimal number

Suppose the given decimal number is 34500.

Step 2: Transforming to a number between 1 and 10

First, adjust the decimal point to locate it after the first significant figure. Here, it would become 3.4500.

Step 3: Counting the number of decimal places moved

In this case, the decimal place has been moved 4 places to the left. This will be important when deciding what power of 10 to multiply by.

Step 4: Constructing the scientific notation

The final step is to multiply the number obtained in Step 2 by \(10^n\), where 'n' is the number of decimal places from step 3. The direction of the decimal movement determines the sign of the number 'n'. If the original number is larger than 1, 'n' is positive and if it’s less than 1, it is negative. Therefore, the number 34500 in scientific notation is written as \(3.4500 × 10^4\).