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Chapter 10: Problem 388

In the following exercises, write each number in scientific notation. $$0.036$$

Short Answer

Expert verified
3.6 \times 10^{-2}

Step by step solution

01

Identify the Decimal Point

Locate the decimal point in the number. In this case, the decimal point in 0.036 is located after the zero.
02

Move the Decimal Point

Move the decimal point to the right until only one non-zero digit remains to the left of the decimal. For 0.036, move it two places to the right to get 3.6.
03

Count the Moves

Count how many places the decimal point was moved. Here, the decimal point moves 2 places.
04

Determine the Exponent

The exponent in the scientific notation is the negative of the number of places moved since we moved the decimal to the right. Thus, the exponent is -2.
05

Write in Scientific Notation

Combine the number and the exponent in the form: \(3.6 \times 10^{-2}\)

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Key Concepts

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

Understanding the Decimal Point
A decimal point separates the whole number part from the fractional part of a number. In the number 0.036, the decimal point is located after the zero.
This indicates that the value is less than one. Comprehending the decimal point's location is crucial for converting numbers into scientific notation.
In scientific notation, we aim to make the number manageable by placing the decimal point after the first significant digit.
Steps to Move the Decimal
Moving the decimal is an essential part of scientific notation. You need to shift the decimal point so that only one non-zero digit remains on its left side.
In the case of 0.036, you move the decimal two places to the right: 0.036 becomes 3.6.
The goal is to create a new number between 1 and 10, which simplifies calculations and representation.
The number of places the decimal point moves will help determine the exponent in the scientific notation.
Negative Exponent Explained
In scientific notation, exponents denote how many times the decimal point has been moved.
If you move the decimal to the right, you use a negative exponent. For 0.036, you moved the decimal two places to the right, thus the exponent is -2.
Negative exponents indicate very small numbers. Understanding this helps in differentiating between large and small values in scientific notation.
Combining the Number and Exponent
After identifying the new number and the exponent, you combine them to complete the scientific notation form.
For 0.036, moving the decimal made it 3.6, and since you moved the decimal two places to the right, the exponent is -2.
Therefore, the scientific notation is written as: \[3.6 \times 10^{-2}\]
This format makes it easier to work with extremely large or small numbers, providing a standardized way to express them.

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