Problem 65 - Algebra and Trigonometry

Question

Write each number in decimal notation. $$ 8.55 \times 10^{-3} $$

Step-by-Step Solution

Answer

0.00855

Step 1: Understand Scientific Notation

Scientific notation expresses numbers as a product of a coefficient and a power of 10. The number given is in the form \( 8.55 \times 10^{-3} \), which means move the decimal point 3 places to the left.

Step 2: Apply the Negative Exponent

Since the exponent is \(-3\), move the decimal point of the number 8.55 three places to the left to get 0.00855.

Step 3: Write the Result in Decimal Notation

After moving the decimal point, the number becomes 0.00855, which is the same number expressed in decimal notation.

Key Concepts

Decimal Notation

Decimal notation is a system of writing numbers where each digit represents a power of ten. It’s the most common way to represent numbers in everyday life. This includes both whole numbers and fractions expressed as decimals. For instance, 128 and 0.75 are both in decimal notation. Unlike scientific notation, which involves a coefficient and a power of ten, decimal notation allows you to see at a glance the value of each digit based on its position.

In more detail, the number 5 in the decimal 25.3 represents 5 tens or 50, while the 3 represents 3 tenths or 0.3. Decimal notation is easy to read and understand, making it ideal for practical use in calculations and day-to-day transactions.

Negative Exponent

A negative exponent in scientific notation indicates that the decimal point is moved to the left, making the number smaller. This is essential when you're working with very small numbers. In the expression \( 10^{-3} \), the exponent \-3\ tells us to move the decimal point three places to the left, transforming a larger number into a smaller one.

For example:

• \( 8.55 \times 10^{-3} \) means the decimal moves left three spaces, resulting in 0.00855.
• \( 4.2 \times 10^{-2} \) becomes 0.042.

Negative exponents make it straightforward to convert back and forth between scientific and decimal notation, especially for very small values.

Power of Ten

The term 'power of ten' refers to expressing a number as ten raised to an exponent. This exponent tells us how many times to multiply 10 by itself. In scientific notation, numbers are written as multiples of powers of ten, which makes it easy to handle very large or very small numbers efficiently.

For example:

• \( 10^3 = 1000 \)
• \( 10^{-1} = 0.1 \)
• \( 10^{-3} = 0.001 \)

Using powers of ten helps simplify calculations and makes it easy to compare the magnitude of different numbers. When you see a number like \( 8.55 \times 10^{-3} \), the \( 10^{-3} \) part shifts the decimal point three places to the left, accurately capturing the number's small size.