Problem 67 - College Algebra and Calculus: An Applied Approach

Question

Use a calculator to evaluate each expression. Write your answer in scientific notation. (Round to three decimal places.)(a) \(0.000345(8,900,000,000)\) (b) \(\frac{67,000,000+93,000,000}{0.0052}\)

Step-by-Step Solution

Answer

The results in scientific notation are: (a) \(3.066 \times 10^{9}\) (b) \(3.077 \times 10^{10}\)

Step 1: Calculate the multiplication in scientific notation

For part (a), the task consists of multiplying \(0.000345\) and \(8,900,000,000\). Using a calculator, the answer can be computed directly, but it needs to be written in scientific notation and rounded to three decimal places.

Step 2: Write the result in scientific notation

The result of the multiplication computes to \(3,065,500,000\), which, when written in scientific notation, becomes \(3.066 \times 10^{9}\).

Step 3: Calculate the division in scientific notation

For part (b), the task consists of adding \(67,000,000\) and \(93,000,000\), and then dividing the sum by \(0.0052\). Again, a calculator can be used to compute the result, but the answer must be converted into scientific notation.

Step 4: Write the result in scientific notation

The result of the addition and subsequent division is calculated as \(30,769,230,769.23077\). Written in scientific notation and rounded to three decimal places, this becomes \(3.077 \times 10^{10}\).

Key Concepts

Calculator Usage

Using a calculator can make complex arithmetic much simpler, especially when dealing with large or small numbers like in scientific notation. Here are some simple steps:

• Enter the first number or expression.
• Use the appropriate function for multiplication or division.
• Enter the second number or value you are calculating with.
• Execute the operation by pressing the equals "=" button.

For scientific notation, calculators can often be set to display results in this format, or you may need to manually convert the standard result to scientific notation. This is especially useful when dealing with extensive numbers like in our example.

Multiplication

In mathematical terms, multiplication is one of the basic arithmetic operations that combines two numbers, called factors, to produce a product. Using a calculator makes it faster when working with large numbers.
Let's consider part (a) of the exercise: we multiply the small number (0.000345) by the large number (8,900,000,000). Inputting each into your calculator and performing the multiplication yields the product. After obtaining the result, convert it to scientific notation to make it more concise. Ensure to check that your calculator is set to the correct number of decimal places for more precise results.

Division

Division is the process of splitting a number into equal parts or groups. It is one of the fundamental operations in arithmetic, just like multiplication. When using a calculator for division, follow these general steps:

• Input the dividend (the number to be divided).
• Press the division symbol (÷).
• Input the divisor (the number you are dividing by).
• Press the equals "=" button to see the result.

Take part (b) of the exercise: first add 67,000,000 and 93,000,000 to get the total, and then divide by 0.0052. The result will often be a very large number, so scientific notation will be helpful.

Rounding Decimal Places

Rounding is crucial when presenting a number that has an extensive amount of digits. It simplifies numbers and makes them easier to understand without losing much precision. Typically, rounding involves adjusting the value of a number to a specified degree of accuracy.
In our example, we are asked to round to three decimal places:

• Look at the fourth decimal place.
• If it's 5 or more, round up the third decimal place by one.
• If it's less than 5, leave the third decimal place as it is.

Rounding helps in achieving a clean conversion to scientific notation without unnecessary complexity in the digits.