Problem 31 - Algebra 1

Question

SCIENTIFIC NOTATION Rewrite in scientific notation. $$ 0.008551 $$

Step-by-Step Solution

Answer

0.008551 in scientific notation is \(8.551 \times 10^{-3}\).

Step 1: Identify the Decimal Point Movement

For the given number 0.008551, make sure the base is between 1 and 10. The first number in the decimal that isn't zero is 8 which is the first digit of the new number. Therefore, the base of the number in scientific notation is 8.551. To determine the exponent, count how many places the decimal point was moved. Here, it was moved three places to the right.

Step 2: Write the Result in Scientific Notation

Now, write the number using the base and the exponent of ten. The base is 8.551 and because the number was moved three places to the right, the exponent will be -3. Thus, 0.008551 in scientific notation is \(8.551 \times 10^{-3}\).

Key Concepts

Decimal Point Movement

When working with scientific notation, the first step is understanding how to move the decimal point correctly. This process, known as decimal point movement, helps place the number you are working with within a suitable range, generally between 1 and 10. For example, let's take the number 0.008551. Here, the decimal point needs to be moved right until it passes the first significant digit, which is 8, forming the number 8.551.

• Identify the first non-zero numeral.
• Move the decimal point until the number is between 1 and 10.
• Count every move as this will affect the exponent.

By moving the decimal three places to the right, we convert 0.008551 into 8.551. Remember, every step moved affects the scientific notation's exponent and indicates the size reduction or increase you're maintaining.

Exponent

In scientific notation, the exponent indicates how many places the decimal point was shifted. It essentially serves as a power of ten.
For a number like 0.008551, which originally had no integer part, moving the decimal right changed it to 8.551.

• Each movement of the decimal point to the right increases the negative exponent by 1.
• For every step to the left, the exponent becomes more positive.
• Hence, in 0.008551, moving three integer spaces to the right results in an exponent of -3.

This results in the scientific notation being written as 8.551 multiplied by 10 raised to the power of -3, or \(8.551 \times 10^{-3}\). This notation simplifies large or minute numbers, making them more manageable and easier to understand.

Base 10

The concept of base 10 is fundamental when it comes to understanding scientific notation. It's the numbering system most commonly used and is based on ten digits: 0 through 9. In scientific notation, every move of the decimal transforms the number by a power of 10. This is why the number is paired with an exponent, as powers of 10 transform the expression.

• Base 10 merely means the mathematical system everyone generally uses in everyday counting and arithmetic.
• Scientific notation uses base 10 to express very large or tiny numbers efficiently.
• For instance, the number 0.008551 becomes \(8.551 \times 10^{-3}\) in scientific notation.

Using base 10, we can easily multiply or divide numbers by altering exponents. This makes calculations less cumbersome, especially in scientific and engineering applications, where precision is key.