Problem 48 - College Algebra

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 Scientific Notation

Scientific notation is a way to write numbers that are too large or too small to be conveniently written in decimal form. It is written as the product of a number between 1 and 10 and a power of 10.

Step 2: Identify the Components of Scientific Notation

In the GDP value given, \( \$ 1.71496 \times 10^{13} \), \(1.71496\) is the coefficient and \(10^{13}\) is the power of ten.

Step 3: Convert Scientific Notation to Standard Notation

To convert from scientific notation to standard notation, move the decimal point in the coefficient to the right by the number indicated by the exponent. Here, move the decimal point 13 places to the right in \(1.71496\).

Step 4: Write the GDP in Standard Notation

Start with the number \(1.71496\) and move the decimal point 13 places to the right, which gives: \(17149600000000\).

Step 5: Final Verification

Double-check the placement of the decimal point to ensure it has moved 13 positions to the right. The final output should be: \(17149600000000\).

Key Concepts

Gross Domestic Product

Gross Domestic Product, abbreviated as GDP, is an important economic metric used to gauge the health of a country's economy. It represents the total monetary value of all goods and services produced over a specific time period within a nation's borders. In this exercise, we looked at the GDP of the United States for the first quarter of 2014.

• It helps in comparing the economic performance of different countries.
• A higher GDP indicates a larger economy and better standard of living.
• GDP can be reported in different intervals like quarterly or annually.

In the exercise, the GDP was initially given in scientific notation as \$1.71496 \times 10^{13}$$. This reflects the vastness of the economy by simplifying extremely large numbers into a more manageable form.

Standard Notation

Standard notation is the typical way of writing numbers where the entire number is written out. Unlike scientific notation, it does not include exponents. This is the form that most people are familiar with when looking at everyday numbers.

• Standard notation is clear and straightforward, avoiding mathematical symbols.
• It's often used in general communications where clear interpretation of numbers is needed.

In the exercise, the GDP was converted from scientific notation \(1.71496 \times 10^{13}\) to standard notation, resulting in \$17,149,600,000,000$$. This is the complete form that avoids any power of ten or coefficients, presenting the full number with all digits visible.

Power of Ten

The power of ten in scientific notation indicates how many times the base number (the coefficient) should be multiplied by 10. It essentially describes how many places the decimal point moves, which can either be to the right (for positive exponents) or left (for negative exponents).

• A positive exponent indicates a large number (e.g., \(10^{13}\) means move the decimal right by 13 places).
• A negative exponent indicates a small number (e.g., \(10^{-3}\) would move the decimal left by 3 places).

For the GDP exercise, the exponent of \(10^{13}\) shows that our decimal in \(1.71496\) is moved 13 spaces to the right, which ultimately provides the enormous figure of dollars in full.

Coefficient

In scientific notation, the coefficient is the number that is multiplied by the power of ten. It is a number between 1 and 10 that simplifies how we express very large or small numbers.

• The coefficient helps retain the significant digits of a number while allowing for easier handling.
• For consistency in scientific notation, it should be a single digit followed by decimals.

In this problem's context, the coefficient is \(1.71496\), which captures the significant digits of the GDP. When the coefficient is combined with the \(10^{13}\), it represents the full economic value in a concise manner, showing just how vast a number the GDP actually is before conversion back to standard notation.