Problem 84 - College Algebra

Question

Write each number in scientific notation. 0.0083

Step-by-Step Solution

Answer

The scientific notation of 0.0083 is \(8.3 \times 10^{-3}\)

Step 1: Write the given number

Start with the number to be converted into scientific notation which is 0.0083

Step 2: Move the Decimal Point.

Move the decimal point in the number so there is only one non-zero digit to the left of the decimal. Here, the decimal place needs to be moved 3 places to the right to get 8.3. Make sure to remember the number of places shifted, as this becomes the exponent of 10 in the scientific notation.

Step 3: Write the Scientific Notation.

Applying the principles of scientific notation, the new value is represented as \(a \times 10^n\). Here, a is the new number 8.3 and n is the number of places the decimal point was moved, which was -3 (moves to the right are negative, to the left are positive). So, the scientific notation of 0.0083 is \(8.3 \times 10^{-3}\).

Key Concepts

Decimal Point

The decimal point is a small but mighty symbol in mathematics. It separates whole numbers from their fractional components. In the number 0.0083, the decimal point appears before the digits, indicating that the value is a fraction of a whole number. This placement shows how much smaller the number is compared to 1. When working with scientific notation, the position of the decimal point is pivotal. Scientific notation requires that we have only one non-zero digit to the left of the decimal point. By adjusting the decimal point position, you can transform the number's appearance without changing its value. For 0.0083, you move the decimal three places to the right to get 8.3. This change makes the number ready for conversion into scientific notation. Remember, each movement is crucial as it defines the number's exponent in the scientific notation equation.

Exponent

Exponents are key in expressing numbers in scientific notation. They convey the decimal point's shift when writing a number concisely. The exponent in scientific notation tells us how many places and in which direction the decimal point moved.When you moved 0.0083's decimal point three places to the right, the exponent turned out to be -3. In scientific notation, moving the decimal point to the right results in a negative exponent, whereas moving it to the left gives a positive exponent. This means:

• Right direction move: Negative exponent
• Left direction move: Positive exponent

By understanding exponents in this context, you can easily determine how to convert any number into scientific notation. Thus, in scientific notation, 0.0083 becomes \(8.3 \times 10^{-3}\), succinctly combining the decimal adjustment with the exponent.

Number Conversion

Converting numbers into scientific notation is a useful skill when handling very large or very small numbers. Scientific notation simplifies these numbers into a format easier to read and understand. When converting numbers like 0.0083, the goal is to express them using a single digit followed by the appropriate power of ten.The conversion process involves several steps, as outlined in our exercise solution. Start by identifying the placement of the decimal and deciding where it should move so only one non-zero digit is left of it. Then, calculate the exponent based on the direction and number of moves made:

• Move the decimal point step-by-step
• Account for the shift by adjusting the exponent
• Finalize the notation form: \(a \times 10^n\)

The final form combines the adjusted decimal with the exponent, making it quick to identify the number's magnitude and precision. For instance, \(8.3 \times 10^{-3}\) directly shows that 0.0083 is a fraction of 1, highlighting the efficiency of scientific notation in managing diverse numerical values.