Problem 103 - 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. $$\frac{480,000,000,000}{0.00012}$$

Step-by-Step Solution

Answer

The answer is \(4 \times 10^{15}\).

Step 1: Division

Divide \(480,000,000,000\) by \(0.00012\). The result of this operation is \(4,000,000,000,000,000\).

Step 2: Conversion to Scientific Notation

Convert this number into scientific notation by moving the decimal point to the left until there is only one nonzero digit to the left of the decimal point, and keep count of the number of places moved. The result of this step is \(4\) with the decimal point moved 15 places, which gives us \(4 \times 10^{15}\).

Step 3: Rounding

The last step involves rounding the decimal factor in the scientific notation answer to two decimal places. In this case, it is not necessary as the decimal factor is already \(4\), which is less than 10 and greater than or equal to 1.

Key Concepts

Division

Division is an important arithmetic operation that involves determining how many times a number (the divisor) is contained within another number (the dividend). In our example, we divided 480,000,000,000 by 0.00012. To solve such division problems efficiently, especially with very large or very small numbers, it can be useful to deal with each number in a way that expresses its magnitude clearly. Thus, a good starting point is to simplify these numbers to make them more manageable, often by expressing them in simpler terms or through scientific notation.

• Here, dividing the given numbers straightforwardly, we obtain 4,000,000,000,000,000 (or 4 quadrillion).
• This process can initially seem intimidating due to the sheer size of the numbers, but breaking it down step by step makes it more approachable.

Separation of the division into these manageable steps ensures clarity in computation.

Decimal to Scientific Notation

Scientific notation is a method of writing very large or small numbers in a compact form, which is especially useful in scientific and engineering fields. This notation can help make calculations easier and more understandable. To convert a number to scientific notation, follow these steps:

• Identify the first non-zero digit in the number.
• Move the decimal point so that only one non-zero digit remains to its left.
• Count the number of places the decimal has moved; this number becomes the exponent of 10.

For example, after dividing, we obtained 4,000,000,000,000,000, which is converted to scientific notation by:- Placing the decimal after the 4, resulting in "4."- The decimal point has moved 15 places to the left, hence the exponent is 15.So, the number in scientific notation becomes \(4 \times 10^{15}\). This method of conversion streamlines the expression of numbers, preserving accuracy while simplifying computation.

Rounding Numbers

Rounding numbers is a common practice used to simplify results, making them easier to work with or report. It's especially crucial when precision is not necessarily paramount, or when dealing with estimates in large datasets. Here's how rounding works:

• Identify the digit to round to. If rounding to two decimal places, focus on the third digit to the right of the decimal.
• If the third digit is 5 or higher, round the second digit up. Otherwise, leave it unchanged.

In our example, after converting to scientific notation, we had \(4 \times 10^{15}\). This number doesn't require any rounding because "4" is already at one significant digit with no digits to round off. Rounding becomes significant in cases where the decimal factor has more figures (e.g., 4.567 becoming 4.57). It's vital to maintain a balance between simplicity and precision, aligning the rounding with the context of its use.