Question

When 2891 is written in scientific notation, the exponent indicating the power of 10 is ________.

Short answer

When 2891 is written in scientific notation, the exponent indicating the power of 10 is 3.

Step-by-step solution

Understanding Scientific Notation

Scientific notation is a method of writing numbers in the format: \(a \times 10^n\), where \(1 \leq a < 10\) and \(n\) is an integer (positive or negative). For example, for the number 1230, in scientific notation, it could be written as \(1.23 \times 10^3\), where \(a = 1.23\) and \(n = 3\). Now let's apply this method to our given number, 2891.

Find the Power of 10

We need to find how many times we need to move the decimal point to the left until we have a number between 1 and 10. Originally, the decimal point is at the end of 2891: \(2891.0\) 1. Move the decimal point one place to the left: \(289.10\) 2. Move the decimal point one more place to the left: \(28.910\) 3. Move the decimal point one more place to the left: \(2.8910\) Now we have found that, after moving the decimal point three times to the left, we get a number between 1 and 10: \(2.8910\).

Write the Number in Scientific Notation

Now that we have moved the decimal point three places to the left, we can put the number in scientific notation form. The number \(2891\) in scientific notation is: \(2.8910 \times 10^3\), where \(a = 2.8910\) and the exponent indicating the power of 10 is \(n = 3\).

Final Answer

When 2891 is written in scientific notation, the exponent indicating the power of 10 is 3.

Key concepts

Power of 10

The power of 10 is a fundamental concept that helps us represent numbers succinctly, especially very large or small ones. In scientific notation, the power of 10 is employed to scale a number, making it more manageable. This representation involves a base number multiplied by 10 raised to an integer exponent.
The base number, often ranging from 1 to less than 10 in absolute value (\(a\) in scientific notation \(a \times 10^n\)), can be adjusted by shifting the decimal point, while the power of 10, denoted by \(n\), compensates for these changes.
For example:

• 10 is represented as \(1 \times 10^1\).
• 100 is written as \(1 \times 10^2\).
• 0.01 is represented as \(1 \times 10^{-2}\).

Each shift of the decimal point to the left increases the power of 10, and each shift to the right decreases it. This allows us to express both standard numbers and their magnitude efficiently.

Decimal Point Relocation

Decimal point relocation is an essential step when converting a number into scientific notation. The aim is to re-position the decimal point so that the number falls between 1 and 10.
Consider the number 2891. Initially, it is understood to be 2891.0, with the decimal at the end. By shifting the decimal point three places to the left, we obtain 2.8910, a value that satisfies the scientific notation condition \(1 \leq a < 10\).
Each move of the decimal point alters the exponent in the power of 10. In our case:

• First move changes 2891.0 to 289.10.
• Second move results in 28.910.
• The third move gives us 2.8910, the proper coefficient for scientific notation.

Therefore, we arrive at \(2.8910 \times 10^3\). The number of places the decimal was moved determines the power of 10 needed, ensuring the original number's magnitude remains unchanged.

Representing Numbers in Mathematics

Numbers can appear in different forms, and representing them economically is a core aspect of mathematics. Scientific notation is particularly useful for handling extremely large or small numbers, providing a clear and consistent way to present them.
When representing numbers in scientific notation, we aim to simplify the reading and calculation processes by concentrating on:

• The significand or mantissa: This is the base number \(a\) in the format \(a \times 10^n\), and it must be a number between 1 and 10.
• Exponent of 10: This indicates how many times the decimal point was relocated. A positive exponent means the original number is large, while a negative exponent means it's small.

For instance, a vast number like 7,000,000 becomes \(7 \times 10^6\), and a tiny number like 0.00056 is \(5.6 \times 10^{-4}\). This method not only simplifies notation but also facilitates easier computation, comparison, and understanding across different scales.