Problem 50 - College Algebra with Corequisite Support

Question

The value of the services sector of the U.S. economy in the first quarter of 2012 was \(\$ 10,633.6\) billion. Rewrite this amount in scientific notation.

Step-by-Step Solution

Answer

The services sector value is \( 1.06336 \times 10^{13} \) in scientific notation.

Step 1: Understanding Scientific Notation

Scientific notation is a way to express numbers that are too large or too small in a concise form. It is written as \( a \times 10^n \) where \( 1 \leq a < 10 \) and \( n \) is an integer.

Step 2: Adjust the Original Number

The number \( 10,633.6 \) billion can be written as \( 10,633,600,000,000 \). We need to express this number in the form of a single digit (\( a \)) followed by the appropriate power of 10.

Step 3: Find the Coefficient \( a \)

To convert \( 10,633,600,000,000 \) into scientific notation, place the decimal after the first non-zero digit to create a number between 1 and 10. This gives us \( 1.06336 \).

Step 4: Determine Power of 10

Count the number of places the decimal has moved from the original position (which was after the last zero in the number \( 10,633,600,000,000 \)). The decimal is moved 13 places to the left, so the power of 10 is 13.

Step 5: Write the Scientific Notation

Combine the coefficient \( a = 1.06336 \) with the power of 10 determined in Step 4. This gives us \( 1.06336 \times 10^{13} \).

Key Concepts

Understanding Coefficient

The term "coefficient" in scientific notation is a critical component that helps simplify large or small numbers. When you express a number in scientific notation, the coefficient is the number that appears first and must be a value between 1 and 10. For instance, in our example with the U.S. economy's services sector value of $10,633.6 billion, the scientific notation involves finding this coefficient. To find the coefficient:

• Place the decimal after the first non-zero digit from the left.
• Continue until you achieve a number between 1 and 10.

In our specific case, $10,633,600,000,000 became 1.06336. Thus, 1.06336 is the coefficient. This figure represents the significant figures of the original number, essentially capturing its size but in a simplified form.

Determining Power of 10

The power of 10 in scientific notation represents how many decimal places the original number's position has been adjusted. This aspect captures the scale of the number, dictating whether it is large or small.For our exercise, the original number $10,633,600,000,000 had the decimal point moved 13 places to the left. Therefore, the power of 10 here is 13.Key points to remember:

• The power is positive if the original number is greater than 1.
• The number of moved places equals the power.

Hence, combining the coefficient (1.06336) with the power of 13 results in scientific notation as 1.06336 \( \times 10^{13} \). This simplifies the interpretation and handling of very large numbers.

The Role of Services in the U.S. Economy

The U.S. economy's services sector plays a pivotal role, being a significant part of its overall structure. This sector includes various industries such as finance, healthcare, education, and retail, which collectively contribute to gross domestic product (GDP).Understanding its economic significance:

• In 2012, services constituted a major portion of the GDP, illustrating immense economic productivity.
• It provides substantial employment opportunities across the nation.

This sector's massive size, like the reported $10,633.6 billion, is often reported in scientific notation to manage large figures conveniently, reflecting both its importance and complexity. Expressing this value in scientific notation as 1.06336 \( \times 10^{13} \) underscores its magnitude in the economy, facilitating a clearer understanding for economic analysis and comparison.