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Chapter 1: Problem 42

Convert the following numbers to standard notation: a. \(5.34 \times 10^{8}\) b. \(4.1 \times 10^{3}\) c. \(6.24 \times 10^{-5}\)

Short Answer

Expert verified
a. 534,000,000 b. 4,100 c. 0.0000624

Step by step solution

01

Understanding scientific notation

Scientific notation expresses numbers as a product of a coefficient and a power of 10. Example: For the expression \(a \times 10^{b}\), \(a\) is the coefficient and \(b\) is the exponent.
02

Convert \(5.34 \times 10^{8}\) to standard notation

To convert \(5.34 \times 10^{8}\): Move the decimal point 8 places to the right because the exponent is positive. \(5.34 \times 10^{8} = 534,000,000\)
03

Convert \(4.1 \times 10^{3}\) to standard notation

To convert \(4.1 \times 10^{3}\): Move the decimal point 3 places to the right because the exponent is positive. \(4.1 \times 10^{3} = 4,100\)
04

Convert \(6.24 \times 10^{-5}\) to standard notation

To convert \(6.24 \times 10^{-5}\): Move the decimal point 5 places to the left because the exponent is negative. \(6.24 \times 10^{-5} = 0.0000624\)

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Key Concepts

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

Understanding Standard Notation
When dealing with scientific notation, we can easily convert numbers into a form that is more compact and manageable. This is particularly useful for very large or very small numbers. In standard notation, numbers are written as they are, without the multiplier of a power of 10. For example:

  • The number \(5.34 \times 10^{8}\)) can be expressed in standard notation as 534,000,000
  • The number \(4.1 \times 10^{3}\)) can be expressed in standard notation as 4,100
  • The number \(6.24 \times 10^{-5}\)) can be expressed in standard notation as 0.0000624

Standard notation helps in visualizing the actual size of the number, making it easier to understand and work with. To convert from scientific notation to standard notation, you simply adjust the decimal point based on the exponent.
Positive Exponent
When the exponent in scientific notation is positive, you move the decimal point to the right to convert it to standard notation. The number of places you move the decimal point is equal to the value of the exponent.

For example:
  • If the exponent is 8 in \(5.34 \times 10^{8}\))
  • You move the decimal point 8 places to the right
This gives us the number 534,000,000.

Similarly:
  • If the exponent is 3 in \(4.1 \times 10^{3}\))
  • You move the decimal point 3 places to the right
This gives us the number 4,100. Notice how increasing the exponent increases the size of the number. This is because each step to the right increases the value of the number by a factor of 10.
Negative Exponent
When the exponent in scientific notation is negative, you move the decimal point to the left to convert it to standard notation. The number of places you move the decimal point is equal to the absolute value of the exponent.

For example:
  • If the exponent is -5 in \(6.24 \times 10^{-5}\))
  • You move the decimal point 5 places to the left
This gives us the number 0.0000624.

Negative exponents indicate very small numbers. Moving the decimal to the left decreases the size of the number by a factor of 10 for each step. This can make large numbers appear much smaller and easier to work with in scientific notation.

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Most popular questions from this chapter

Astrology makes testable predictions. For example, it predicts that the horoscope for your star sign on any day should fit you better than do horoscopes for other star signs. Read the daily horoscopes for all of the astrological signs in a newspaper or online. How many of them might fit the day you had yesterday? Repeat the experiment every day for a week and keep a record of which horoscopes fit your day each day. Was your horoscope sign consistently the best description of your experiences?

The following astronomical events led to the formation of life on Earth- including you. Place them in order of their occurrence over astronomical time. a. Stars died and distributed heavy elements into the space between the stars. b. Hydrogen and helium were made in the Big Bang. c. Enriched dust and gas gathered into clouds in interstellar space. d. Stars were born, and they processed light elements into heavier ones. e. The Sun and planets formed from a cloud of interstellar dust and gas.

The surface area of a sphere is proportional to the square of its radius. How many times larger is the surface area if the radius is a. doubled? b. tripled? c. halved (divided by 2 )? d. divided by 3 ?

A light-year is a unit of distance most comparable to the distance from Earth to the a. Moon. b. Sun. c. outer Solar System. d. nearest star. e. nearest galaxy.

If a car is traveling at \(35 \mathrm{km} / \mathrm{h}\), how far does it travel in a. one hour? b. half an hour? c. one minute?
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