Problem 106 - Algebra and Trigonometry

Question

Perform the indicated computations. Write the answers in scientific notation. If necessary, round the decimal factor in your scientific notation answer to two decimal places. $$ \frac{66,000 \times 0.001}{0.003 \times 0.002} $$

Step-by-Step Solution

Answer

The answer expressed as a decimal in scientific notation is \(1.1\times10^{7}\)

Step 1: Perform the multiplication

First, perform the multiplication of \(66000 \times 0.001\) and keep the result aside. The result is \(66\). Similarly, multiply \(0.003 \times 0.002\), which is \(6\times10^{-6}\).

Step 2: Perform the Division

Next, divide the result of the first multiplication by the result of the second multiplication. Hence, \(66/(6\times10^{-6})\) becomes \(11\times10^{6}\) in preliminary scientific notation.

Step 3: Convert to Scientific Notation

The preliminary scientific notation can now be converted into real one. The scientific notation for an answer is written as one digit in the ones place, a decimal, and the remaining digits, all multiplied by \(10^n\), where \(n\) is an integer. This results in \(1.1\times10^{7}\).

Key Concepts

Multiplication and Division

When it comes to operations involving scientific notation, it is crucial to master multiplication and division. Let's take a look at how to tackle this. The first step is to manage each of the numbers separately and simplify where possible.

For multiplication, the straightforward method is to multiply the first set of numbers given in a problem, as we see when multiplying large numbers like 66,000 by much smaller numbers such as 0.001. This results in a manageable number, 66. For division, you take the results of the multiplication from two different parts and divide one by the other. This is exactly what happens when dividing 66 by 0.003 times 0.002, already presented as scientific notation in our solution as 6 times 10 to the power of negative 6. The key is to handle coefficients and powers of ten separately.

• Multiply or divide the base numbers first.
• Handle the exponents next.
• If multiplying, add the exponents; if dividing, subtract them.

This process leads us step by step to a simplified number, prepared for conversion into its scientific form.

Rounding Decimals

When using scientific notation, the decimal number often needs to be rounded for simplicity and clarity. Precise rounding helps present data in its most useful form. Consider the multiplication of 66,000 by 0.001, resulting in 66. When division comes into play, this number needs to be converted as part of a quotient. Arriving at an answer sometimes requires rounding.

Here’s a quick guide for rounding decimals in scientific notation:

• Identify how many decimal places you need (like two decimal places in our example).
• Look at the number in the slightest decimal place beyond what you need.
• If it is 5 or greater, round up the last needed decimal place.
• If it's less than 5, keep the last needed place unchanged.

Once you complete the division in our example, you might arrive at interim scientific notation results requiring adjustment like rounding 11 to 1.1. Adjust using the rounding process for a tidy figure suitable for reporting.

Conversion to Scientific Notation

Scientific notation is a method for expressing numbers that are too big or too small in a simple, concise format. After performing multiplication and division, like in the problem we solved, it's essential to convert the result into proper scientific notation. This simplifies calculations and improves readability.

To convert a number to scientific notation, remember these steps:

• Place the decimal after the first non-zero digit.
• Count the number of places you moved the decimal. This becomes the power of 10.
• If you moved the decimal to the left, the exponent is positive; if to the right, the exponent is negative.

In our specific example, resulting in 11 times 10 to the power of 6, we formatted it to 1.1 times 10 to the power of 7 to ensure proper scientific notation. This way, numbers remain simple to read and work with, avoiding cumbersome digits. Converting such calculations accurately is pivotal to handling larger or more complex numbers in scientific notation efficiently.