Problem 48 - College Algebra with Corequisite Support

Question

The Gross Domestic Product (GDP) for the United States in the first quarter of 2014 was \(\$ 1.71496 \times 10^{13}\). Rewrite the GDP in standard notation.

Step-by-Step Solution

Answer

The GDP in standard notation is 17,149,600,000,000.

Step 1: Understanding the Scientific Notation

The GDP is given in scientific notation as \(1.71496 \times 10^{13}\). Scientific notation expresses numbers as a product of a number between 1 and 10 and a power of 10.

Step 2: Interpret the Exponent

The exponent 13 indicates the decimal point should move 13 places to the right. This is because the exponent is positive, which means the number is very large.

Step 3: Rewrite the Number

Starting with 1.71496, moving the decimal point 13 places to the right fills the additional places with zeros. After moving the decimal, the sequence becomes 17,149,600,000,000.

Key Concepts

Gross Domestic Product (GDP)

Gross Domestic Product, commonly known as GDP, is a crucial economic indicator used worldwide to gauge the economic health of a country. It represents the total dollar value of all goods and services produced over a specific time period within a country's borders.
Understanding GDP helps economists and policymakers determine whether an economy is growing or shrinking. Here's why GDP is essential:

• Economic Size: GDP provides an overview of the market value of all finished goods and services produced.
• Growth Measurement: It indicates the economic growth or decline by comparing the GDP of different time periods.
• Policy Making: It assists in making informed decisions regarding fiscal and monetary policies.
• International Comparison: GDP allows comparison of economic productivity between different countries.

For the United States, in the first quarter of 2014, the GDP was given as \(1.71496 \times 10^{13}\). Interpreting such numbers commonly involves scientific notation, making it easier to communicate and handle large numbers.

Exponent Interpretation

Interpreting exponents is a fundamental skill in understanding scientific notation. An exponent indicates how many times a number, the base, is multiplied by itself. When dealing with scientific notation, the exponent will tell us how many times the decimal point should move.

• Positive Exponent: A positive exponent, as seen in GDP \(1.71496 \times 10^{13}\), means you move the decimal point to the right. This suggests a large number.
• Negative Exponent: A negative exponent means the decimal point moves to the left, typically representing a small number.

In our example, an exponent of 13 shifts the decimal 13 positions to the right, emphasizing the vast scale of the number. Hence, the number becomes significantly larger in standard form.Understanding exponents in scientific notation thus makes large or small numerical expressions more manageable and easier to work with.

Standard Notation

Standard notation is the most common way of writing numbers without exponents. This is what you typically see in everyday economic reports or balance sheets.

• This format involves displaying all digits without the multiplication of powers of ten.
• It’s particularly useful for reading and interpreting economic data, like GDP, as it offers a straightforward representation of value.

To convert from scientific notation to standard notation, follow these steps:1. Identify the decimal point in the given number.2. Move the decimal point as many times as indicated by the exponent.3. Fill in any empty places with zeros.In our example of \(1.71496 \times 10^{13}\), moving the decimal 13 places right gives us the GDP figure in standard notation: 17,149,600,000,000. This allows for an easier understanding of the economic data's magnitude without exponents.