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Chapter 5: Problem 95

Convert to scientific notation. $$ 407,000,000,000 $$

Short Answer

Expert verified
4.07 \times 10^{11}

Step by step solution

01

Identify the significant figures

Find the non-zero digits in the number. Here, 407,000,000,000 has significant figures 407.
02

Place the decimal point

Rewrite the significant figures with a decimal point after the first digit: 4.07.
03

Count the number of places moved

Count how many places the decimal has moved from its original place to the new place (after the first digit). The decimal point in 407,000,000,000 moves 11 places to the left to become 4.07.
04

Express in scientific notation

Combine the earlier steps to express the number in scientific notation: \[ 4.07 \times 10^{11} \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Significant Figures
Significant figures represent the digits in a number that contribute to its accuracy and precision. Any non-zero digit is considered a significant figure. In the number given in the exercise, 407,000,000,000, the significant figures are 4, 0, and 7, making up '407'. Zeroes between non-zero digits are also significant. Remember, significant figures help in maintaining the precision of a number during calculations. For instance, suppose we're converting the number 407,000,000,000 to scientific notation. We must identify the significant figures first because these core digits will be used to form the accurate representation of the number.
Decimal Point
The decimal point is crucial when converting a number to scientific notation. It separates the integer part from the fractional part of a number. When dealing with large numbers, moving the decimal point helps simplify the expression. In the exercise, we took the significant figures '407' and placed the decimal point after the first digit, turning it into '4.07'. This step is vital because it standardizes the significant figures, preparing the number for the final scientific notation form. Moving the decimal point effectively transforms a large or small number into a manageable form that is easy to read and use in calculations.
Place Value
The place value system assigns value to digits based on their position in a number. Each movement of the decimal point changes the value by a factor of 10. For example, in our number, 407,000,000,000, each position shift to the left (for each '0') increases the value by 10 times. To convert it to scientific notation, we count the number of places the decimal moves: 11 places to the left. Thus, 407,000,000,000 becomes 4.07 脳 1011. Understanding place value ensures accurate conversion and helps in grasping how large or small a number actually is. It also maintains the integrity of the initial number's size through scientific notation.

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