Problem 54 - Beginning and Intermediate Algebra

Question

Convert to scientific notation. $$0.0056$$

Step-by-Step Solution

Answer

The scientific notation for \(0.0056\) is \(5.6 \times 10^{-3}\).

Step 1: Determine the value of a

To find the value of \(a\), we need to move the decimal point in the given number until we get a number between 1 and 10. Keep track of the number of places the decimal point has been moved. 0.0056 can be rewritten as: \(5.6 \times 10^{-3}\) Here, \(a=5.6\).

Step 2: Determine the value of b

The value of \(b\) represents the number of places the decimal point has been moved. As we moved the decimal point 3 places to the right, the value of \(b\) is -3. Remember, when you move the decimal point to the right, the exponent is negative, and when you move it to the left, the exponent is positive.

Step 3: Write the number in scientific notation

Now we have the values of \(a\) and \(b\), which are 5.6 and -3, respectively. We can now write the number in scientific notation: \(5.6 \times 10^{-3}\) So, the scientific notation for 0.0056 is \(5.6 \times 10^{-3}\).

Key Concepts

Decimal Point Movement

When converting numbers to scientific notation, moving the decimal point is the first important step. This involves shifting the decimal to form a new number between 1 and 10.
For instance, consider the number 0.0056. To change this into scientific notation, you need to move the decimal point, so the number becomes 5.6.

• If you move the decimal point to the right, observe how many times you shift it to reach the new format.
• In this case, shifting it three places to the right means you multiply by a power of ten.
• Movement to the left, however, implies a negative exponent, as seen with basic decimal adjustments.

Think of decimal point movement as a trick to control the size of your number efficiently. The number of times you move it directly impacts the exponent you will apply later.

Exponent Rules

Exponent rules are vital in scientific notation since they determine the power of ten that accompanies your new number. Here, the direction of your decimal shift truly matters.

• Moving the decimal point to the right, as noted in our example, results in a negative exponent.
• Think of the number 0.0056. We moved it three spaces right to form 5.6, so we write it as \(10^{-3}\).
• Conversely, moving left indicates a positive increase in the power of ten.

These rules allow you to translate any decimal movement directly to a corresponding mathematical exponent. Emphasizing these helps ensure accuracy in calculations and communications, especially in scientific settings.

Numbers Between 1 and 10

Numbers between 1 and 10 are essentially the core building blocks of scientific notation. In order to apply this form, the leading number, or coefficient, must always fall within this range.

• Strive for simplicity and standardization; moving your decimal point until you achieve a number like 5.6 is crucial.
• This ensures consistency, allowing others to understand and compare numbers easily.
• Being between 1 and 10 means the integer part is significant, with decimal portions illustrating the precision of your measurement.

By ensuring your value fits between these bounds, you're setting up a universally recognizable scientific entry. It’s a tiny detail but fundamentally influences all calculations following this conversion.