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Chapter 2: Problem 30

Write the following numbers in scientific notation. a. 0.0006730 b. 50000.0

Short Answer

Expert verified
a. \(6.73 \times 10^{-4}\) \n b. \(5 \times 10^{4}\)

Step by step solution

01

- For number less than 1

Shift the decimal point to the right until you have a number between 1 and 10. In 0.0006730, shifting the decimal point 4 places to the right gives 6.73. The number of places you shifted the decimal point is the value of the power of 10. Since the original number was less than 1, this will be a negative power. The scientific notation of 0.0006730 is therefore written as \(6.73 \times 10^{-4}\).
02

- For number greater than 1

Shift the decimal point to the left until you have a number between 1 and 10. In 50000.0, shift the decimal point 4 places to the left to get 5. The number of places you shifted the decimal point is the value of the positive power of 10. Thus, 50000.0 in scientific notation is written as \(5 \times 10^{4}\).

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Key Concepts

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

Understanding the Decimal Point in Scientific Notation
The decimal point is a crucial element when converting numbers to scientific notation. By shifting the decimal point, you adjust the size of the number to fall between 1 and 10, making it easier to handle, especially when dealing with very large or tiny numbers.
There are two main scenarios when shifting the decimal point:
  • **For numbers less than 1**: Move the decimal point to the right. This process increases the number to be between 1 and 10 and decreases its power of ten with a negative exponent. Shifting 0.0006730 four places right gives 6.73.
  • **For numbers greater than 1**: Move the decimal point to the left. This reduces the large number to fit between 1 and 10, increasing its power of ten with a positive exponent. For instance, shifting 50000.0 four places left results in 5.
Remember that the number of shifts determines the exponent's magnitude in scientific notation. It's a simple yet effective method to express numbers compactly.
Navigating Powers of 10 in Scientific Notation
Powers of 10 are the backbone of scientific notation, forming part of the number's representation. A power of 10 indicates how many times you multiply the number '10' by itself. Examples include:
  • **Positive powers**: Indicate multiplication by 10. For example, a power of 4 means 10 x 10 x 10 x 10.
  • **Negative powers**: Indicate division or multiplication by a fraction, essentially dividing 1 by the power of 10. For example, a power of -4 implies 1 divided by 10, four times (1/10,000).
Understanding powers of 10 allows you to quickly determine the magnitude change when shifting the decimal point. In scientific notation, like in our examples, a positive power of 10 grows the number significantly, while a negative power reduces it.
Exploring Exponents in Scientific Notation
Exponents are the numerical expression's superscripts in scientific notation, indicating how many times the base (usually 10) is used in multiplication. When writing numbers in scientific notation, exponents are crucial components. Here's how it works:
  • **Positive exponents**: Show that the base number (10) has been multiplied rather than divided. This occurs when the original number is greater than 1, hence shifted left: e.g., for 50000.0, the exponent is 4.
  • **Negative exponents**: This occurs when the number is originally less than 1, hence shifted right. It means dividing by the base number: e.g., for 0.0006730, the exponent is -4.
Grasping the role of exponents in scientific notation is beneficial since they illuminate how far the number deviates from its basic form, influencing its size based on how many times we've scaled it up or down with powers of 10.

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