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Chapter 1: Problem 24

Convert to decimal notation. $$3 \times 10^{-6}$$

Short Answer

Expert verified
0.000003

Step by step solution

01

Understand Scientific Notation

Scientific notation is a way to express very large or very small numbers. It has the format of a number between 1 and 10 multiplied by a power of 10. In this case, we have the number 3 multiplied by 10 raised to the power of -6.
02

Interpret the Exponent

The exponent -6 indicates that the decimal point needs to be moved 6 places to the left. This is because a negative exponent means the number is very small.
03

Move the Decimal Point

Starting with 3 (which can be written as 3.0), move the decimal point 6 places to the left. Each place you move the decimal point adds a zero.
04

Write the Result in Decimal Notation

After moving the decimal point 6 places to the left, 3 becomes 0.000003. This is the decimal notation of the number.

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Key Concepts

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

decimal notation
To understand decimal notation, we need to know how numbers are expressed in a standard numerical format. Decimal notation is the way we write numbers using digits 0-9 and a decimal point. For instance, the number 3,000,000 can be written in decimal form as 3 followed by six zeros. In contrast, very small numbers use zeros in front of them, such as 0.000003 for three millionths. This is crucial when converting numbers from scientific notation to decimal notation to ensure accuracy.
negative exponent
In scientific notation, exponents tell us how many times to multiply or divide by 10. A negative exponent means the number is very small and indicates the decimal point must be moved to the left. For example, in the expression $$3 \times 10^{-6}$$, the exponent -6 instructs us to move the decimal point 6 places to the left. So, converting this expression to decimal notation, we shift the decimal from 3 (equivalent to 3.0) six places to the left, resulting in 0.000003. This helps in dealing with very tiny values in a manageable format.
moving decimal point
Converting from scientific to decimal notation involves moving the decimal point according to the power of 10. Here are the steps with an example:
  • Start with the number: 3
  • Since it's expressed as $$3 \times 10^{-6}$$, we identify the exponent -6

The exponent -6 tells us to move the decimal point 6 places to the left. Starting from 3.0, we move it across six digits, filling in zeros as placeholders: 0.000003. Practicing this movement will help solidify the concept of transitioning between scientific and decimal notation effectively.

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Most popular questions from this chapter

Calculate using the rules for order of operations. If an expression is undefined, state this. $$ \frac{8 \div 4 \cdot 6\left|4^{2}-5^{2}\right|}{9-4+11-4^{2}} $$

Write an equivalent expression using the distributive law. $$ 9 a(b-c+d) $$

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Write an equivalent expression using the distributive law. $$ 5 x(y-z+w) $$

Calculate using the rules for order of operations. Use the commutative, associative, and distributive laws to show that \(5(a+b c)\) is equivalent to \(c(b \cdot 5)+a \cdot 5 .\) Use only one law in each step of your work.
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