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Chapter 0: Problem 79

Write each number in scientific notation. $$638,000,000,000,000,000$$

Short Answer

Expert verified
The scientific notation for 638,000,000,000,000,000 is \(6.38 \times 10^{18}\).

Step by step solution

01

Identifying the number

The given number is 638,000,000,000,000,000.
02

Determining digit a and exponent n in the scientific notation

Move the decimal point of the origin number such that only one non-zero digit remains on the left. This digit will be 'a' and the moved step will be the factor '10^n', where 'n' is the number of steps the decimal point was moved to the left. In the case of this number, we need to move the decimal 18 places to the left to obtain 6.38, so a = 6.38 and n = 18.
03

Writing the scientific notation

Thus, in scientific notation, this number can be written as \(6.38 \times 10^{18}\).

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Key Concepts

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

Understanding the Decimal Point
The decimal point is a small dot used in mathematics to separate the whole number part from the fractional part in a decimal number. When writing large numbers, the position of the decimal point is crucial, especially when converting a number into scientific notation. By moving the decimal point, we can simplify how a number is expressed.
  • To convert a large number into scientific notation, the decimal point is shifted so that there is only one non-zero digit to its left.
  • For 638,000,000,000,000,000, the decimal point is moved 18 places to the left, transforming it into 6.38.
  • This single digit on the left (6.38) is essential as it represents the 'significant figures' of the number.
This manipulation using the decimal point makes the number easier to handle and understand, especially in scientific and engineering contexts.
The Power of Exponentiation
Exponentiation is a mathematical operation that involves a base number raised to the power of an exponent. It is a shorthand way of expressing repeated multiplication of a base by itself. In scientific notation, exponentiation plays a central role as it helps express how many times the decimal point has been moved.
  • When converting a number to scientific notation, you determine how many places the decimal point has been moved. This count becomes the exponent.
  • In our example, the decimal was moved 18 places to the left, creating the expression \(10^{18}\).
  • The exponent, 18, indicates not only the scale of the number but also provides a concise way to express very large or small quantities powerfully and neatly.
Exponentiation simplifies mathematical expressions and calculations, making it possible to handle large numbers with ease.
Handling Large Numbers
Large numbers can appear daunting, but scientific notation offers a way to make them more approachable. This number form uses the concepts of decimal points and exponentiation to express large amounts succinctly.
  • Rather than writing out 638,000,000,000,000,000, we use the format \(a \times 10^n\).
  • Through scientific notation, this number becomes \(6.38 \times 10^{18}\).
  • This transformation not only saves space and time but also minimizes potential errors when reading or performing calculations.
Using scientific notation, large numbers are condensed into a manageable size, allowing clearer communication and better comprehension, particularly in fields like astronomy and physics where vast quantities frequently occur.

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