Problem 9 - Algebra for College Students

Question

Write each of the following in scientific notation. For example \(27800=(2.78)(10)^{4}\). \(376.4\)

Step-by-Step Solution

Answer

\(3.764 \times 10^2\)

Step 1: Identify the Significant Figures

The non-zero digits in the number 376.4 are '3764'. To prepare these for scientific notation, express it as 3.764 by placing the decimal point after the first non-zero digit.

Step 2: Determine the Power of Ten

Count how many places the decimal point was moved to get from 376.4 to 3.764. Here, the decimal point was moved 2 places to the left. Each move to the left increases the power of ten by 1, so we have an exponent of 2.

Step 3: Construct the Scientific Notation

Combine the significant figure (3.764) with the power of ten (10). Thus, 376.4 in scientific notation is written as \(3.764 \times 10^2\).

Key Concepts

Significant Figures

Significant figures are the digits in a number that contribute to its precision. These include all non-zero numbers, any zeros between non-zero digits, and any trailing zeros in the decimal part. Recognizing significant figures helps to retain the number's accuracy when it's expressed in different forms like scientific notation.
For example, in the number 376.4, the digits '3', '7', '6', and '4' are the significant figures. When converting to scientific notation, these figures determine the initial value, which becomes 3.764, ensuring the precision of the original measurement is maintained.
It's important to handle significant figures properly in calculation to avoid misleading results. They represent what is known with confidence, leaving the rest for approximation.

Powers of Ten

Scientific notation heavily relies on powers of ten. It offers a convenient way to express extremely large or small numbers. Here, numbers are written as a product of a significant figure and ten raised to an exponent.
In scientific notation, the powers of ten indicate how many times you multiply the base number by 10. Moving the decimal point to the right or left affects the power of ten used in the notation.
With 376.4, when expressed in scientific notation as 3.764, the decimal point shift to the left by two places gives the power of ten an exponent of 2. Therefore, it is written as \(3.764 \times 10^2\). This method allows for a more compact representation of numbers, particularly in scientific calculations.

Decimal Point Movement

Understanding decimal point movement is crucial when working with scientific notation. The decimal point's position change tells you about the power of ten.
For the number 376.4, the decimal is originally after the last digit '4'. When converting the number to scientific notation, the decimal moves to create a number between 1 and 10, which is 3.764 in this case.
You moved the decimal two places to the left, increasing the exponent of the power of ten to 2, indicating \(10^2\). Each place moved to the left increases the exponent since it means the number is larger than one. Conversely, moving it right would decrease the power of ten, indicating a smaller number. Remembering this pattern simplifies converting any number to scientific notation.