Problem 13 - Algebra for College Students

Question

Write each of the following in scientific notation. For example \(27800=(2.78)(10)^{4}\). \(0.0214\)

Step-by-Step Solution

Answer

0.0214 is \(2.14 \times 10^{-2}\) in scientific notation.

Step 1: Identify the Decimal Shift

To write the number in scientific notation, we need to express it as a number between 1 and 10, multiplied by a power of 10. First, we identify how many places we need to shift the decimal to the right to get a number between 1 and 10. For the number 0.0214, we shift the decimal two places to the right, which gives us 2.14.

Step 2: Determine the Power of 10

Since we shifted the decimal two places to the right, we need to balance this by multiplying by a power of 10. Each shift to the right corresponds to a negative power of 10 because the original number, 0.0214, was less than 1. Therefore, we multiply by \(10^{-2}\) to account for the two shifts.

Step 3: Write in Scientific Notation

Combine the number obtained in Step 1 with the power of 10 from Step 2. Thus, 0.0214 can be written in scientific notation as \(2.14 \times 10^{-2}\).

Key Concepts

Decimal Shift

Writing a number in scientific notation often begins with the decimal shift. The objective is to transform the number into a quantity between 1 and 10. This step requires moving the decimal point to a new position. For example, to express the number 0.0214 in scientific notation, we move the decimal point two places to the right. This process gives us the number 2.14, which is neatly positioned between 1 and 10.
Why do we do this? It's simple! A core goal in scientific notation is to maintain a standard format that makes numbers universally comparable. Shifting the decimal ensures we start with a clear, concise number that sets up the other essential parts of scientific notation effortlessly.

Power of 10

After shifting the decimal, we must consider the power of 10. This shows the magnitude of the decimal shift. In scientific notation, this is represented as an exponent in base ten notation.
Every move of the decimal corresponds to a power of 10:

• Each move to the right decreases the exponent by 1.
• Each move to the left increases the exponent by 1.

In our example with 0.0214, the decimal was shifted two places to the right. This movement yields a negative exponent because the original number is less than 1. Therefore, it results in a power of \(-2\). This power precisely articulates how many places the decimal was shifted compared to the original number.

Negative Exponent

A negative exponent is a key aspect when dealing with numbers less than one in scientific notation. It tells us how many times to divide by 10, or how many decimal places were moved to the right during the decimal shift.
In the case of 0.0214, the number was initially below 1, prompting a rightward shift for the decimal. Shifting two spaces to the right equates to a \(10^{-2}\) in the notation.
The exponent being negative essentially states, "Take this base number, and divide it by 100," because \(10^{-2} = 1/100\). This balances the new position of the decimal, ensuring that the scientific notation accurately and succinctly represents the magnitude of the original number. Every negative exponent thus becomes a blueprint for reconstructing the original value from the scientific notation.