Problem 102 - College Algebra

Question

Perform the indicated computations. Write the answers in scientifi c notation. If necessary, round the decimal factor in your scientific notation answer to two decimal places. $$\frac{1.5 \times 10^{-2}}{3 \times 10^{-6}}$$

Step-by-Step Solution

Answer

The answer is \(5.0 \times 10^{3}\)

Step 1: Divide the Decimal Factors

Divide 1.5 by 3 to get 0.5.

Step 2: Subtract the Exponents

Subtract the exponent of denominator which is -6 from the exponent of numerator which is -2 to get 4 as \(10^{4}\)

Step 3: Write in Scientific Notation

Combine the divided decimal factors and subtracted exponent to write the final answer in scientific notation. So, the answer is \(0.5 \times 10^{4}\)

Step 4: Adjust Decimal Point and Exponent

To make sure that only one digit appears before the decimal, adjust the decimal point and exponent. It will become \(5.0 \times 10^{3}\)

Key Concepts

Decimal Factors

The first step in solving problems involving scientific notation is often dealing with decimal factors. Decimal factors refer to the numbers that precede the exponent portion of a scientific notation. In our original exercise, these factors are 1.5 and 3.
To handle them, we perform the standard arithmetic operation, which in this case is division. When you divide these decimal factors, you divide the numerator by the denominator:

• For the given example, you calculate \(\frac{1.5}{3}\), which results in 0.5.

Working with decimal factors is straightforward because it relies on basic division rules. But it's crucial to ensure accuracy, especially if you're dealing with more complex factors that require rounding.
Rounding is key when your division results in long decimals. For scientific notation, precision is essential, and often you'll round your decimal factor to two decimal places.

Exponents

Exponents in scientific notation define the power of ten by which your decimal factor is multiplied. Learning to manipulate exponents is fundamental to mastering scientific notation.
In scientific notation, when you divide two numbers, you subtract the exponent in the denominator from the exponent in the numerator.
For our problem, you start with exponents as follows:

• The numerator has an exponent of -2 (from \(10^{-2}\)).
• The denominator has an exponent of -6 (from \(10^{-6}\)).

To find the exponent of the result, perform the subtraction: \[-2 - (-6) = 4\].
Thus, converting the division to an exponential expression leaves you with \(10^{4}\). This is a standard approach when dividing exponential expressions, and understanding this step is crucial for performing more complex scientific notation arithmetic.

Division in Scientific Notation

Division in scientific notation involves simplifying both the decimal factors and the exponents before combining them into a final expression. Understanding this process helps you manage complex calculations with large or small numbers easily.
Once you've worked out the division of decimal factors and adjusted the exponents, the remaining process involves recomposing these into a proper scientific notation format.

• In our example, after dividing the decimals and adjusting exponents, you combine the results: 0.5 times \(10^{4}\).
• However, scientific notation requires that only one non-zero digit appears to the left of the decimal. To achieve this, adjust the placement of the decimal point.
• Shift the decimal in 0.5 to become 5.0, which involves altering the exponent.

So, the final scientific notation becomes: \(5.0 \times 10^{3}\).
This adjustment ensures the number fits the standard layout of scientific notation and makes it easier to read and compare with other similarly formatted numbers. Knowing how to manipulate both the decimal factor and the exponent refinements are what make scientific notation so powerful and convenient for handling large numerical calculations.