Problem 76 - Algebra A Combined Function

Question

Write each number in scientific notation. $$ 0.00000017 $$

Step-by-Step Solution

Answer

0.00000017 in scientific notation is \(1.7 \times 10^{-8}\).

Step 1: Identify the Decimal Position

The number given is 0.00000017. The first step is to find the position of the first non-zero digit, which is '1' at the eighth position after the decimal point.

Step 2: Form the Standard Form Expression

Move the decimal point 8 places to the right to form the number 1.7. This is because 1.7 is the form used for the base number in scientific notation, that is, a number between 1 and 10.

Step 3: Determine the Power of Ten

Since we moved the decimal 8 places to the right, this means we are multiplying by 10 to the negative power of 8: \(10^{-8}\). This negative exponent indicates the initial decimal movement to the right in a number less than 1.

Step 4: Combine Form and Power

Combine the base number and the power of ten to express the original number. Therefore, the number 0.00000017 in scientific notation is \(1.7 \times 10^{-8}\).

Key Concepts

Decimal Movement

When dealing with very small numbers, like 0.00000017, understanding decimal movement is crucial to converting them into scientific notation. The goal is to adjust the decimal point so that the resulting number is between 1 and 10. This involves identifying the first non-zero digit in the decimal number.

In this example, 0.00000017, the first non-zero digit is '1', which appears in the eighth position to the right of the decimal point. By moving the decimal 8 places to the right, we transform it into 1.7. This is essential because the base number in scientific notation must always be any number ranging from 1 to 10, inclusive. Understanding this movement of decimals is the first step to mastering scientific notation.

Negative Exponent

In scientific notation, the exponent on the 10 indicates how many places the decimal point has been moved. For numbers less than 1, like 0.00000017, we encounter a negative exponent. This is because the decimal point has moved to the right to reach a position where the base number is between 1 and 10.

For this number, we moved the decimal 8 places to the right to create the number 1.7. This movement is represented by the power of ten: \(10^{-8}\). A negative exponent simply informs us that the number originally was a small fraction, requiring decimal adjustment to the right.

Standard Form Expression

A standard form expression in scientific notation succinctly expresses both very large and very small numbers. It consists of two parts:

• The coefficient (base number) which should be between 1 and 10.
• The base of ten raised to an appropriate exponent which shows the decimal movement.

To convert 0.00000017 into scientific notation, you begin with decimal movement, then you determine the correct exponent. The result, 1.7, represents the base, because it falls between 1 and 10.

The final scientific notation expression becomes \(1.7 \times 10^{-8}\). This not only conveys the value but also offers a simple way to handle extremely small numbers efficiently in calculations and communication.