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Chapter 7: Problem 22

Convert each number into scientific notation. $$0.0000002$$

Short Answer

Expert verified
2 \times 10^{-7}

Step by step solution

01

Identify the Base Number

The base number is the significant digits of the given number excluding the leading zeros. In this case, the base number is 2.
02

Count the Decimal Places

Count the number of decimal places the decimal point needs to be moved to place it just after the first non-zero digit. Here, the decimal point needs to be moved 7 places to the right.
03

Formulate the Scientific Notation

Combine the base number with the power of 10. Since the decimal point was moved to the right, the exponent is negative. Therefore, the scientific notation for 0.0000002 is written as: $$2 \times 10^{-7}$$

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Base Number
In scientific notation, identifying the base number is the first crucial step. The base number consists of the significant digits of the given number, excluding any leading zeros. For the example given in the exercise, the number is 0.0000002. Here, the significant digit is 2.
The base number provides the core value of the number without considering its scale. It must be a number between 1 and 10 (excluding 10), also known as the *normalized fraction*.
By isolating the base number, you make the subsequent steps simpler and keep the scientific notation process systematic.
Decimal Places
After identifying the base number, counting the decimal places is the next key step. This helps to determine the exponent in the scientific notation.
You count how many places the decimal point needs to be shifted to transform the original number into the base number. In the exercise example of 0.0000002, the decimal point is moved 7 places to the right to position it immediately after the first significant digit, which is 2.

A useful tip while counting:
  • When moving the decimal to the right, the exponent will be negative.
  • When moving the decimal to the left, the exponent will be positive.
Accurate counting ensures the scientific notation correctly reflects the original value.
Exponent
Formulating the exponent in scientific notation finalizes the process. The exponent indicates the number of decimal places the point was moved. In the example exercise, moving the decimal 7 places to the right results in a negative exponent. Therefore, you write: \(2 \times 10^{-7}\).
The exponent in scientific notation serves to express very large or small numbers efficiently. Here are some benefits:/ul>
  • It simplifies comparing and computing significantly different scales of numbers.
  • Provides a concise format for displaying large data, useful in sciences like physics and chemistry.
    • The exponent complements the base number to maintain the exact value in a more manageable form. By learning to handle the exponent effectively, you master scientific notation easily.

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