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Chapter 5: Problem 87

Convert to decimal notation. $$ 3.497 \times 10^{-6} $$

Short Answer

Expert verified
0.000003497

Step by step solution

01

Understand Scientific Notation

Scientific notation is a way to write very large or very small numbers. It is written as the product of a number between 1 and 10 and a power of 10. In this problem, the number given is \(3.497 \times 10^{-6}\).
02

Interpret the Exponent

The exponent of \(-6\) indicates that the decimal point should be moved \(6\) places to the left. A negative exponent means the number is very small, less than 1.
03

Move the Decimal

Start with the number \(3.497\). To move the decimal point 6 places to the left, write zeros to fill in the spaces. \(3.497\) becomes \(0.000003497\).
04

Write the Final Answer

After moving the decimal point, we can see the number in decimal notation is \(0.000003497\).

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Key Concepts

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

Decimal Notation
Decimal notation is the standard way of writing numbers using digits 0 to 9, and placing a decimal point to represent fractions of whole numbers. This is the most familiar way for us to represent numbers in everyday life. For example, the number 1.5 means 1 whole and 5 tenths.

When converting from scientific notation to decimal notation, the goal is to write the number as a straightforward series of digits. In our exercise, we started with the scientific notation form of a number, which is a compact way to represent it. By transforming it step by step, we end up with an easy-to-read decimal format.
Negative Exponent
A negative exponent in scientific notation tells us how many places to move the decimal point to the left. A negative exponent essentially means dealing with a very small number. Think of the exponent as indicating the number of divisions by 10.

In the example given:

  • For the number 3.497 x 10^{-6}, the exponent is -6.
  • This means the decimal point moves 6 places to the left.
So, a negative exponent shrinks the number significantly and helps us manage and work with very small values conveniently. It's useful in scientific fields where extremely small measurements or quantities are often required.
Moving Decimal Point
Moving the decimal point is the key action you take when converting scientific notation to decimal notation. You adjust the position of the decimal point based on the exponent in the scientific notation.

For a negative exponent like -6, you move the decimal point left, adding zeros as necessary.
Let's walk through it step by step:

  • Start with the number 3.497.
  • The exponent is -6, so move the decimal point 6 places left.
  • The number changes to 0.000003497.
The movement of the decimal point is crucial to accurately converting between notations, ensuring the values remain equal and correctly represented in decimal form.

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