Problem 17 - Algebra for College Students

Question

Write each of the following in scientific notation. For example \(27800=(2.78)(10)^{4}\). \(0.00000000194\)

Step-by-Step Solution

Answer

The scientific notation is \(1.94 \times 10^{-9}\).

Step 1: Identify the Significant Figures

Look at the number provided: \(0.00000000194\). The significant figures are 194.

Step 2: Determine the Exponent by Counting Decimal Places

To convert the number into scientific notation, count how many places you move the decimal to the right until you reach the first significant figure. Here, you move the decimal 9 places to the right: \(1.94\).

Step 3: Write in Scientific Notation

Combine the significant figure with the power of ten, reflecting the decimal shift. Since the original number was less than 1, the exponent is negative: \(1.94 \times 10^{-9}\).

Key Concepts

Significant Figures

Significant figures are crucial for representing the precision of a number. In any given number, significant figures include all the digits starting from the first non-zero digit. They help indicate the reliability of a measurement by showing how exact it is. In the example, the number is given as \(0.00000000194\). Here the significant figures are '194'. This is because all the digits after removing the leading zeros (before the first non-zero digit) are counted as significant figures.

• Leading zeros aren't counted as significant figures as they only serve as placeholders.
• Non-zero digits are always considered significant.
• Zeros between non-zero digits are also significant.

Identifying significant figures correctly ensures accuracy when converting numbers to scientific notation.

Decimal Places

Decimal places are the numbers located after the decimal point in a number. Counting decimal places closely is essential, especially when converting numbers to scientific notation. In the exercise, the task was to identify the number of decimal places the decimal point would move. For the number \(0.00000000194\), you move the decimal point 9 places to the right, reaching the first significant figure '1'.

• The objective is to place the decimal right after the first significant figure.
• Each movement of the decimal signifies a base10 factor shift, which will help form the exponent in the scientific notation.

By counting the decimal places correctly, you can derive the appropriate exponent necessary to express the number in scientific notation.

Exponent

An exponent in scientific notation is a power of ten used to express the number's decimal place movement. This approach simplifies large or small numbers, making them easier to read and write. The exponent indicates how many times the base number (10) is multiplied by itself. In this case, moving the decimal 9 places to the right gives the exponent as \(-9\), since the original number \(0.00000000194\) is less than one.

• A positive exponent signifies that the original number was greater than one.
• A negative exponent indicates the original number was a decimal smaller than one.
• The size of the exponent directly corresponds to the decimal place shifts required.

In scientific notation, the number is represented compactly as \(1.94 \times 10^{-9}\). Understanding exponents is key to mastering scientific notation, reflecting the magnitude efficiently.