Problem 42 - Elementary and Intermediate Algebra

Question

Write number in scientific notation. \(7,007,000,000\)

Step-by-Step Solution

Answer

The number is written in scientific notation as \( 7.007 \times 10^9 \).

Step 1: Identify the Significant Figures

The number provided is 7,007,000,000. To write this in scientific notation, we first identify the significant figures. The significant figures here are 7.007, as these are the non-zero and important figures in the number.

Step 2: Determine the Decimal Placement

To convert to scientific notation, the decimal must be placed after the first non-zero digit. So, we rewrite the significant figures as 7.007.

Step 3: Count the Exponent of 10

Count how many places the decimal must move to the right to return to the original number position. Starting from 7.007 to 7,007,000,000, the decimal moves 9 places to the right.

Step 4: Write in Scientific Notation

Using the significant figures from Step 2 and the exponent from Step 3, we can write the number in scientific notation as: \[ 7.007 \times 10^{9} \]

Key Concepts

Significant Figures

When working with scientific notation, identifying the significant figures is a crucial first step. These are the digits in a number that carry meaningful information about its precision. For the number 7,007,000,000, focus on non-zero digits. Here, the significant figures are 7, 0, 0, and 7. Zeros can also be significant if they are sandwiched between other non-zero digits; they suggest the measured or estimated precision of the number. In our example, the significant figures are 7.007, which includes the non-zero digits and any zeros that provide vital scale information.

Decimal Placement

Placing the decimal correctly is essential to convert a number to scientific notation. This process ensures the number is expressed in a way that highlights its significant figures, simplifying comparisons and calculations. Begin by placing the decimal after the first non-zero digit. In 7,007,000,000, this leads to 7.007. The position holds the significant figures only while maintaining the size of the number. Remember, decimal placement is about making numbers manageable and highlighting key digits without altering their value.

Exponent of 10

The exponent of 10 in scientific notation tells us how many times and in which direction to move the decimal place. It indicates the magnitude of the original number. For 7,007,000,000, once we place the decimal after 7, the decimal has "moved" 9 places to the right. Thus, the exponent becomes 9. This movement, represented as an exponent base 10, means the number has been multiplied by 10 nine times to achieve its original state.In scientific notation, we combine our significant figures with this exponent, writing the number as: \[ 7.007 \times 10^{9} \]