Problem 47 - Elementary and Intermediate Algebra

Question

Write number in scientific notation. 11

Step-by-Step Solution

Answer

11 in scientific notation is \(1.1 \times 10^1\).

Step 1: Identify the Base Number

Scientific notation is expressed as a number between 1 and 10 multiplied by a power of 10. First, we check if the number 11 can be adjusted to fall between 1 and 10. By dividing 11 by 10, we get 1.1, which is within the range of 1 to 10. Hence, 1.1 will be our base number.

Step 2: Determine the Exponent

Next, determine the power of ten needed to convert the base number back to the original number. Since we divided by 10 to get 1.1 (from 11), we multiply by 10^1 to restore the original number, so the exponent is 1.

Step 3: Write in Scientific Notation

Combine the base number and the power of ten into scientific notation format. Therefore, 11 in scientific notation is written as \(1.1 \times 10^1\).

Key Concepts

Base Number

When writing a number in scientific notation, the first step is to identify the base number. The base number is a crucial component of scientific notation. It is essentially a number that is transformed to fall between 1 and 10. Why is this important? Well, scientific notation is a way to simplify and express very large or very small numbers efficiently. By ensuring the base number is between 1 and 10, it makes handling the number easier.

• The base number retains the significant figures of the original number, ensuring we don't lose precision.
• If the original number is greater than 10, we divide by the appropriate power of ten to bring it into the 1 to 10 range.
• If the original number is less than 1, we multiply by the appropriate power of ten.

When dealing with the number 11, we divide by 10, giving us a base number of 1.1. This is perfect as it lies within the ideal range, retaining the original number's prominence while simplifying it.

Power of Ten

The power of ten in scientific notation acts as a multiplier that scales the base number back to the original number’s magnitude. It's literally what allows the incredibly large or tiny numbers to be presented compactly.

• The power of ten is shown as an exponent on 10, indicating how many times 10 is multiplied by itself.
• A positive exponent signals a large number, indicating how many zeros follow the digit if we were to expand it.
• A negative exponent represents a small number, indicating the number of decimal places to the left of the digit.

For example, when 11 is divided by 10, it scales down to 1.1, and to scale it back, we need to multiply 1.1 by 10, which is expressed as a power of ten: \(10^1\). Thus, our power of ten here is 1.

Exponent Calculation

Calculating the exponent is a key part of writing numbers in scientific notation. The exponent shows how many times the base number should be multiplied by the power of ten to return to the original value. This requires some simple manipulation of the number.

• Determine whether to multiply (for numbers greater than 1) or divide (for numbers less than 1) to achieve a base number between 1 and 10.
• The number of times 10 is multiplied by itself becomes the exponent.
• It's crucial; the exponent tells us the scale of the number.

In the example of converting 11 to scientific notation, we want our base number as 1.1. Realizing that this involved dividing 11 by 10 once, we recognize our calculation requires 10 to the power of 1. Hence, the exponent here is 1, indicating a singular shift at a power of ten necessary to return to 11 from 1.1.