Problem 36 - Algebra 2

Question

For Exercises \(36-38,\) use the graph at the right. U.S. growers were forecasted to produce 264 million pounds of pecans in 2003 . Write the number of pounds of pecans forecasted by U.S. growers in 2003 in scientific notation.

Step-by-Step Solution

Answer

264 million is 2.64 x 10^8 in scientific notation.

Step 1: Understand the Problem Statement

We need to convert the given number of pecans, measured in pounds, from standard notation to scientific notation. The number provided is 264 million pounds.

Step 2: Convert Million to Full Number

First, convert 264 million to its full numerical form. Since one million is represented by six zeros, 264 million is written as 264,000,000.

Step 3: Write in Scientific Notation

Scientific notation requires expressing the number as a product of a number between 1 and 10, and a power of 10. The number 264,000,000 can be rewritten as 2.64 multiplied by 10 to the eighth power, i.e., 264,000,000 = 2.64 x 10^8.

Step 4: Verify the Scientific Notation

Ensure there is one non-zero digit to the left of the decimal point. For 264,000,000, the number 2.64 meets this criterion because moving the decimal point 8 places to the right gives us the original number.

Key Concepts

Standard Notation

Standard notation is a way to write numbers that we use in our everyday lives. It means expressing a number in its complete form without using exponents or decimals to make it short. For example, if you have the number 264 million, it is written in standard notation as 264,000,000.
Whenever we write numbers like this:

• The digits before any commas show the hundreds, thousands, and millions.
• The commas, read from left to right, separate thousands and millions for easier readability.

Most of the time, standard notation is perfect for writing numbers we encounter daily. However, for very large or very small numbers, other ways, such as scientific notation, are more convenient.

Powers of 10

Powers of 10 are an essential concept in mathematics, especially for working with large numbers. When we talk about powers of 10, we mean multiplying 10 by itself a certain number of times. For example,

• 10 raised to the power of 1 is 10 (or 10^1).
• 10 raised to the power of 2 is 100 (or 10^2), which is 10 times 10.
• Continuing this pattern, 10 raised to the power of 8, or 10^8, is 100,000,000.

When you use powers of 10, you can efficiently express complex numbers with a simple short form. This is especially useful in scientific notation. In scientific notation, you write a number as a product of a digit list and a power of 10, which keeps everything concise and easy to manage.

Conversion Process

The conversion process from standard notation to scientific notation involves several steps. Let's use the example from the exercise where you have 264,000,000 pounds of pecans.
Here's how you convert that number:

• Identify the digits starting from the left until you form a number between 1 and 10. For 264,000,000, this number is 2.64.
• Count the number of decimal places you have moved to get from 264,000,000 to 2.64. In this instance, you move the decimal eight places to the left.
• Express the original number as a multiplication between the number formed and a power of 10. Here, it's 2.64 times 10 raised to the eighth power, written as 2.64 x 10^8.

This process can be applied to any large or small number, enabling them to be expressed in a condensed and readable form, thanks to scientific notation.